The intuitive criterion

I have asked a similiar question before, but I would very much appreciate if someone would say if my reasoning in this particular case is correct.

Consider the down below: For part a) I have found only one pooling PBE, namely

$$s^{PBE} = ((bike,bike),(no,no))$$

with beliefs that $$\mu_B=1/4$$ and $$\mu_P \leq 1/2$$. Then for part b) my answer is:

The intuitive criterion is a refinement to rule out "unreasonable" off-path beliefs. The belief $$\mu_p$$ that supports the PBE says that the bank thinks it is not too likely that a Porsche driver is a rich customer. According to the intuitive criterion, it would be an unreasonable belief if the poor customer was worse off by deviating to Porsche for any belief and that rich customer was better of for some belief. This is actually the case. For $$1/2 > \mu_P$$, the poor customer would be worse off by deviating to Porsche since the bank would choose yes after observing this signal, i.e. yielding him a payoff off $$-1$$ compared to 0.

Therefore, we can rule out this equlibrium as a possible pooling PBE by the intuitive criterion.

Is this correct?

TIA for any help.