Spanning Tree - 5 GHz Link with 2.4 GHz Backup

I am thinking of a point-to-point link using 5.x GHz bridges,
with 2.4 GHz bridges as an always-on backup:

A1 ( ------------ ) A2
B1 ( ------------ ) B2

A1, A2 are the 802.11a bridges, while B1, B2 are the 802.11b bridges.
A1 and B1 are connected to the same switch at one site.
A2 and B2 are connected to the same switch at the other site.
B1 and B2 support Spanning Tree Protocol (STP) but the A bridges
do not, and neither do any of the switches.

Will this setup work well and as intended? (Namely, when the A link
is up, the B link will be blocked. If the A link fails, the B link
will take over seamlessly. When the A link starts working again,
the B link will again become blocked.)

In other words, is it enough for the B bridges *alone* to support STP?
Or maybe even just one of the B bridges?
(As opposed to all of the bridges, or the switches on either side.)

TIA