We show that the minimax sample complexity for estimating the pseudo-spectral gap $γ_{\mathsf{ps}}$ of an ergodic Markov chain in constant multiplicative error is of the order of $$\tildeΘ\left( \frac{1}{γ_{\mathsf{ps}} π_{\star}} \right),$$ where $π_\star$ is the minimum stationary probability, recovering the known bound in the reversible setting for estimating the absolute spectral gap [Hsu et al., 2019], and resolving an open problem of Wolfer and Kontorovich [2019]. Furthermore, we strengthen the known empirical procedure by making it fully-adaptive to the data, thinning the confidence intervals and reducing the computational complexity. Along the way, we derive new properties of the pseudo-spectral gap and introduce the notion of a reversible dilation of a stochastic matrix.