The longer of your two attached log files, log-network-manager-without-segmentation.txt, seems to contain at least one of your two tests, so I extracted the part from your test starting (last start of network-manager, with log verbosity set to contain <trace> and <debug> messages) up to the end. I have attached this part to this posting as log-network-manager-without-segmentation-2.txt.

When one searches for the string "current configuration:" one finds where DNS configuration is changed due to network interfaces being added or removed. The last of these occurrences is the only place where your VPN (with your office's DNSes 172.27.0.42 and 172.27.0.33) gets added to your phone's internet access (with DNS 192.168.42.129). Only in this acase you get an error with pacrunner (PAC = Proxy Auto Configuration, see https://wiki.gnome.org/Projects/NetworkManager/Proxies) but seems to succeed later. There are also some other errors visible. Here are some lines of the log:

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lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <trace> [1564066391.4886] dns-mgr: current configuration: [{'nameservers': <['172.27.0.42', '172.27.0.33']>, 'interface': <'tun2'>, 'priority': <50>, 'vpn': <true>}, {'nameservers': <['192.168.42.129']>, 'interface': <'enp0s20f0u2'>, 'priority': <100>, 'vpn': <false>}]
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <debug> [1564066391.4895] dispatcher: (32) (tun2) dispatching action 'vpn-up'
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <debug> [1564066391.4900] pacrunner: call[0x563b6899bc30]: send: new config ({'Interface': <'tun2'>, 'Method': <'direct'>, 'BrowserOnly': <false>, 'Domains': <['172.31.252.244/24', '172.27.0.0/24', '172.27.240.0/20', '192.168.222.0/24', '185.48.32.186/32', '80.247.66.0/24', '87.248.32.84/32', '178.239.180.149/32', '80.247.66.96/32', '87.248.32.223/32', '10.6.0.0/16', '172.31.252.0/24']>},)
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <trace> [1564066391.4900] pacrunner: call[0x563b6899bc30]: sending...
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <debug> [1564066391.4907] pacrunner: call[0x563b6899bc30]: sending failed: GDBus.Error:org.freedesktop.DBus.Error.NameHasNoOwner: Name "org.pacrunner" does not exist
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <debug> [1564066391.4907] prope
rties-changed[0x563b688b9c10]: type NMDeviceTun, iface NMDBusDeviceStatisticsSke
leton: {'TxBytes': <uint64 48>}
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <debug> [1564066391.4908] prope
rties-changed[0x563b688b9c10]: type NMDeviceTun, iface NMDBusDeviceTunSkeleton: 
{'Dhcp4Config': <objectpath '/'>, 'Dhcp6Config': <objectpath '/'>, 'Ip4Address':
 <uint32 4110163884>, 'Ip4Config': <objectpath '/'>, 'Ip6Config': <objectpath '/
'>, 'IpInterface': <''>, 'Managed': <true>, 'State': <uint32 20>, 'StateReason': <(uint32 20, uint32 41)>}
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <debug> [1564066391.4909] device[0x563b688b9c10] (tun2): ip4-config: update (commit=0, new-config=0x563b688ab4a0)
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <debug> [1564066391.4910] device[0x563b688b9c10] (tun2): ip6-config: update (commit=0, new-config=0x563b688b4940)
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <debug> [1564066391.4912] manager: (tun2): assume: generated connection 'tun2' (7ddca237-b609-4f44-bf54-9f3705d663ce)
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <debug> [1564066391.4917] settings: Failed to add 7ddca237-b609-4f44-bf54-9f3705d663ce/'tun2': Plugin does not support adding connections
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <debug> [1564066391.4922] settings-connection[0x563b68793a80]: constructed (NMSKeyfileConnection)
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <trace> [1564066391.4929] settings-connection[0x563b68793a80,7ddca237-b609-4f44-bf54-9f3705d663ce]: update settings-connection flags to visible (was none)
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <trace> [1564066391.4929] settings-connection[0x563b68793a80,7ddca237-b609-4f44-bf54-9f3705d663ce]: update settings-connection flags to unsaved,visible (was visible)
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <info>  [1564066391.4929] keyfile: add connection in-memory (7ddca237-b609-4f44-bf54-9f3705d663ce,"tun2")
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <debug> [1564066391.4935] settings-connection[0x563b68793a80,7ddca237-b609-4f44-bf54-9f3705d663ce]: failed to read connection timestamp: Il file chiavi non presenta alcuna chiave «7ddca237-b609-4f44-bf54-9f3705d663ce» nel gruppo «timestamps»
[...]
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <debug> [1564066391.5344] dns-mgr: (device_state_changed): no DNS changes to commit (0)
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <info>  [1564066391.5344] device (tun2): Activation: successful, device activated.
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <debug> [1564066391.5344] device[0x563b688b9c10] (tun2): set metered value 4
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <debug> [1564066391.5345] dispatcher: (34) (tun2) dispatching action 'up'
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <debug> [1564066391.5350] pacrunner: call[0x563b689a10c0]: send: new config ({'Interface': <'tun2'>, 'Method': <'direct'>, 'BrowserOnly': <false>},)
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <trace> [1564066391.5350] pacrunner: call[0x563b689a10c0]: sending...
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <debug> [1564066391.5351] connectivity: (tun2) faking request. Connectivity check disabled
lug 25 16:53:11 CLIFMI085 NetworkManager[21584]: <debug> [1564066391.5351] device[0x563b688b9c10] (tun2): remove_pending_action (0): 'queued-state-change-activated'
[...]
----------

I did not find out which test this is actually. I need logs which can clearly get associated with the test for which they are (with/without segmentation).