Compatibility conjecture for the sandwiched Rényi perspective

Let $n$ be a positive integer, let $\mathbb{H}^n_+$ denote the cone of positive semidefinite Hermitian $n\times n$ matrices, and let $\Psi_\alpha$ be the matrix function defined in the paper. A function is $(K,\nu)$-compatible with a domain when it satisfies the compatibility conditions used for self-concordant barriers.

Compatibility conjecture. For $\alpha\in[2,\infty)$, the function $-\Psi_\alpha$ is $(\mathbb{H}^n_+,(2\alpha-1)/3)$-compatible with the domain $\mathbb{H}^n_+\times\mathbb{H}^n_+$ for any positive integer $n$.

The scalar analogue is proved, and its compatibility parameter is shown to be tight. Numerical experiments suggest that the same lower bound remains tight for matrices of every dimension, but the conjecture is open; it would clarify the construction of efficiently computable self-concordant barriers for the relevant cones.