Suppose $X_1,X_2,\ldots$ is a reversible Markov chain with state space of size $k$ and absolute spectral gap $\gamma_*.$ What is the spectral gap of the (non-reversible) Markov chain $Y_1,Y_2,\ldots,$ where $Y_i = (X_i,X_{i+1})?$
(Spectral gap of a non-reversible chain is defined as, say that of its multiplicative reversibilization.)