Mixed Strategies in Bayes Nash Equilibrium (Bayesian Battle of the Sexes). Shouldn't it depend on $p$?

I have a question about calculating mixed strategies in a Bayes Nash Equilibrium in a simple 2-player bimatrix game. To demonstrate the issue, consider “Bayesian Battle of the Sexes.” Suppose P1 faces a P2 whose type is unknown with probability $p$. An example is below in which $p=.5$.

Here’s what’s surprising to me. If you calculate the mixing strategy for P1 by making sure that P2 is indifferent, you wind up with P1 mixing strategies that don’t depend at all on $p$. This seems strange to me.

If P2’s type is more clear, shouldn’t the optimal mixed strategy shift to anticipate this?

End of question. Addendum: The game above apparently has two mixed strategy EQMs:

1. P1 mixes $(frac{1}{3},frac{2}{3})$, P2 left mixes (0,1), P2 right ($frac{2}{3},frac{1}{3}$).
2. P1 mixes $(frac{2}{3},frac{1}{3})$, P2 left mixes ($frac{2}{3},frac{1}{3}$), P2 right (0,1).

Like I said, this doesn’t seem to depend on $p$ (the probability of facing the first type).

Also: Shouldn’t this game have an odd # of EQM? I thought this was a rule of thumb. I am contemplating the idea that there’s a third equilibrium that mixes the two above.