= Solution
\b[False.] Again use two states, but now take
$$
P=
\begin{pmatrix}1&0\\0&1\end{pmatrix},
\qquad
Q=
\begin{pmatrix}\tfrac12&\tfrac12\\
\tfrac12&\tfrac12
\end{pmatrix}.
$$
The support condition holds. Under $P$, the <first return time> to state one is identically one, so $\mu_1=1$. Under $Q$, the chain is irreducible with <stationary distribution> $(1/2,1/2)$. The <mean recurrence time> formula gives
$$
\eta_1=\frac1{\pi_1}=2>1=\mu_1.
$$
\b[Thus enlarging transition support does not imply a smaller mean return time.]
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