Is that really true for F? I thought BA auto-allocates seat assignments when tickets are issued in F.

For the OP’s question, F capacity has gone from 14 to 8, leading to a lot of flights without A class availability currently, hence the higher prices. That could change as BA could adjust their fare class availability for the PHX route based on various factors, but a quick reduction in capacity has definitely had an impact on F prices on this route thus far.