Audenaert–Datta conjecture on the full joint convexity range

Let $\mathcal M_N$ denote the set of complex $N\times N$ matrices and $\mathcal P_N$ the subset of positive definite matrices. For $p,q\in\mathbb R$, $s>0$, and $K\in\mathcal M_N$, define

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

Audenaert–Datta conjecture. If $1\leq p\leq2$, $-1\leq q<0$, and $s\geq1/(p+q)$, then this map is jointly convex. These conditions complement the necessary conditions stated in the source and conjecturally give the sufficient range for joint convexity.