Are you saying that the ESXi host are reported as disconnected only in the vSphere Web Client whereas the C# client does not report any disconnect?
I would first configure vCenter alarms to send notifications when there is a host disconnect, this will help isolate issues between ESXi/vCenter and C#/Web Client. A host "not responding" indicates that the ESXi host is unable to send/receive heartbeats to vCenter and if I am not correct is UDP port 902. I understand that you have checked networking already but you can ask your network team to check the firewall for TCP/UDP port 902.
In addition to the above, also monitor if there are any tasks that are being initiated on the ESXi host that may cause this disconnect, for e.g. taking snapshots.
Hope this helps !!