Effective uniqueness of the measure of maximal entropy in Ornstein's $\bar{d}$-metric

Every topologically mixing shift of finite type has a unique measure of maximal entropy. It follows from Ornstein theory that every invariant measure with entropy close to maximal must be close to the MME in the $\bar{d}$-metric. We prove a quantitative version of this, strengthening earlier results of Kadyrov that established effective uniqueness in the Wasserstein metric.