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If I understood correctly, you are asking whether the spectral gap $\gamma=1-\rho$ of a non-reversible Markov chain $P$ provides any universal control on the Poincaré constant (which is the spectral gap of the additive reversibilization of $P$, or in your notation, $1-\|P\|$). The answer is no, even on finite state spaces: consider the Markov chain on $\{0,1\}^n$ which, at each step, replaces the current state $x=(x_1,\ldots,x_n)$ with either $(x_2,\ldots,x_{n-1},0)$ or $(x_2,\ldots,x_{n-1},1)$, each with probability a half. This chain has the maximum possible spectral gap, namely $\gamma=1$. Yet, its Poincaré constant tends to $0$ as $n\to\infty$.