Audenaert–Datta conjecture on joint convexity of an α-z trace functional

Let $\mathcal M_N$ denote the set of complex $N\times N$ matrices and $\mathcal P_N$ the subset of positive definite matrices. For $p,q\in\mathbb R$ and $K\in\mathcal M_N$, consider the map

$$\mathcal P_N\times\mathcal P_N\ni(A,B)\mapsto\operatorname{Tr}\left(B^{q/2}K^*A^pKB^{q/2}\right)^{1/(p+q)}.$$

Audenaert–Datta conjecture. If $1\leq p\leq2$ and $-1\leq q<0$, then this map is jointly convex. The conjecture specifies the parameter regime in which the trace functional underlying the $\alpha-z$ Rényi entropies has the required convexity.