Uniqueness of the physical measure for the maps $\varphi_{\alpha,b}$

Let $\varphi_{\alpha,b}$ be one of the maps in the setting of Theorem~; an SRB/physical measure is a measure with the corresponding SRB or physical property. Uniqueness conjecture. There exists at most one SRB/physical measure for each of the maps $\varphi_{\alpha,b}$ in the setting of Theorem~. This is motivated by known uniqueness results for SRB measures of transitive surface diffeomorphisms and for certain non-conservative skew-products with two-dimensional center bundles having mixed behavior, while the conjectured uniqueness remains open for this family.