I have the same issue. Just installed Proxmox to one of my servers a few days ago. Before there was Windows Server running on the same hardware.
I have two Intel NICs: I210 for MGMT traffic and I219-LM for VM data traffic with VLAN tagging.
When I do an backup or move a lot of data between the VMs and an external destination, the VMs loosing connection until I shut/ no shut the port on the switch where the nic is connected.

I downgraded to kernel 6.8.12-8-pve but with no success, now changed the NIC setting with ethtool like mentioned by Hegse and could do the backup without problems.
Tonight I have some sync jobs running, let's see if the connection is broken again tomorrow or not. But for the moment it looks like that helped.

But I just bought a pcie card with dual 2,5G lan and Realtek chipset for this server. I have another Proxmox host with a quad 2,5G realtek pcie card and never had problems with it.