= Solution
Let
$$
m=\max_x\mathbb E_xT_A
$$
and choose $x$ attaining the maximum. For every $t$, the <Strong Markov property> at time $t$ gives
$$
m=\mathbb E_xT_A
\leq t+m\mathbb P_x(T_A>t).
$$
Thus
$$
\mathbb P_x(T_A\leq t)\leq\frac tm.
$$
If $t\geq t_{\mathrm{mix}}(1/4)$, then
$$
\mathbb P_x(X_t\in A)
\geq\pi(A)-\frac14\geq\frac14.
$$
Since $\{X_t\in A\}\subseteq\{T_A\leq t\}$, this is impossible when $t<m/4$. Allowing for integer times, one may take any smaller absolute constant, for example
$$
\boxed{t_{\mathrm{mix}}(1/4)\geq\frac18
\max_x\mathbb E_xT_A.}
$$
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