Sufficiency conjecture for minimal quantum Rényi divergences under positive maps

A dichotomy is a pair $(\rho,\sigma)$ of finite-dimensional density operators. The minimal quantum Rényi divergences, also called the sandwiched quantum Rényi divergences, are denoted by $D^{\min}_\alpha$. Interconversion is by positive, trace-preserving maps.

Minimal Rényi sufficiency conjecture. The family $D^{\alpha}_{\min}$ provides a sufficient family of divergences for interconversion of finite-dimensional density matrices via positive, trace-preserving maps.

All customary quantum Rényi divergence families are known not to be sufficient for interconversion by completely positive trace-preserving maps, while the minimal (sandwiched) family remains a candidate after passing to positive trace-preserving maps. Whether it is sufficient is left open.