Find a separating perfect Bayesian equilibrium Exercise Question 2, Chapter 28, Strategy: An Introduction to Game Theory 3rd Edition by Joel Watson


In part (a) of the question, we have to check if any separating perfect Bayesian equilibrium exists.
Now the solution manual says that there is not any separating perfect Bayesian equilibrium.
My attempt:
The separating strategy profiles are AB’ and BA’.
Let us take AB’. P1 plays B’, then P2 knows that P1 must be L-type. P2 will prefer playing Y, so P1 will not prefer playing B’. No PBE here.
Let us take BA’. P1 plays B then P2 knows that P1 is H-type. P2 prefers playing X, so P1 will not prefer deviating from B since 6 &gt; 4. So, A’B is a separating perfect Bayesian equilibrium here.
What is wrong with my understanding here?
$BA'$ doesn't work because $A'$ is not a best response to $X$, which is a best response to $B$.