Connectedness conjecture for the category of categories with adjunctions

Let $\operatorname{Cat}^{\triangleq}$ be the category whose objects are categories and whose morphisms are the adjunction-based morphisms described in the source. A category is connected when it is inhabited and there is a finite sequence of morphisms between any two of its objects; here this notion is generalized to possibly infinite sequences. Connectedness conjecture. $\operatorname{Cat}^{\triangleq}$ is a connected category. This conjecture asserts global connectedness for the category of categories equipped with the specified morphisms; the source provides no resolution or further evidence.