Can lxd bridge create logical L2 subnet across L3 switches?

My lxd cluster physical hosts are in different L3 switches, there’re 2 vms: vmA (on hostA) and vmB (on hostB):

vmA and vmB are in same subnet (xxx/24), but their hosts (hostA & hostB) are in different L3 switches with different subnet address.

I found vmA and vmB cannot ping each other, by tcpdump and ip neigh, I found it cannot find ARP mac address.

Each cases other than the above are fine if:
vmA & vmB are in different lxd bridge subnet (although host across L3)
vmA & vmB are in the same lxdbr subnet (but their hosts are in the same L3 switch set)

I want to know the reason, can anyone help me? This has been bothering me for a long time.
Did I miss lxd bridge configs? (bgp configured)
Does bridge type network not support logical L2 across physical L3?
Is ovn type network required for this case?

Any help or tips are greatly appreciated!