Strong sequential equilibria and the existence of others
There are three classes of equilibria of this game.

The first class is sequential:
\begin{equation}
(s_1,s_2)=(y,r)
\end{equation}
and the beliefs are
\begin{equation}
\mu_1(a)=\mu_1(b)=\mu_2(a\mid y)=\mu_2(b\mid y)=\frac12.
\end{equation}


The second class is not sequential, but weak perfect Bayesian:
\begin{equation}
(s_1,s_2)=(x,l)
\end{equation}
and the beliefs are
\begin{equation}
\mu_1(a)=\mu_1(b)=\frac12,\quad\text{but }\mu_2(a\mid y)=1-\mu_2(b\mid y)=p&gt;\frac58.
\end{equation}


The third class is actually a boundary case of the second class, which permits P2 to use mixed strategies. This equilibrium requires P2's off equilibrium belief to be
\begin{equation}
\mu_2(a\mid y)=1-\mu_2(b\mid y)=\frac58,
\end{equation}
so that P2 can play a mixed strategy that puts sufficiently high probability on $l$ ($\sigma_2(l)\ge\frac35$). It is also necessary that P1 does not play $y$ with positive probability, because if they do, Bayes rule would kick in, requiring that P2's belief be $(\frac12,\frac12)$ instead of $(\frac58,\frac38)$ which rationalizes P2's mixed strategy.