Assume we have a homogeneous Markov chain with $3$ states $s_1,s_2,s_3$ and the transition probability matrix is
$P = \left[\begin{array}{ccc} \frac{1}{2} & 0 & \frac{1}{2}\\ 0 & \frac{1}{2} & \frac{1}{2}\\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \end{array}\right].$
The steady-state probability is therefore $\left[\begin{array}{ccc} 0.2857 & 0.2857 & 0.4286\end{array}\right].$
Next, I need to switch one of the regular states, say $s_3$, to an absorbing state, and thus the transition probability matrix becomes
$Q = \left[\begin{array}{ccc} \frac{1}{2} & 0 & \frac{1}{2}\\ 0 & \frac{1}{2} & \frac{1}{2}\\ 0 & 0 & 1 \end{array}\right]$
The steady-state probability now is $\left[\begin{array}{ccc} 0 & 0 & 1\end{array}\right].$
The question: is it possible to show, in general, that the mixing time can be reduced by switching one of the regular states to the absorbing state in a homogeneous Markov chain?