Joint convexity of assemblage relative entropy

Let $d,r,s$ be natural numbers, let $n>0$ be a natural number, and let $p_0,\ldots,p_{n-1}$ be strictly positive real numbers with $\sum_{j<n}p_j=1$. For each $j<n$, let $\rho^j,\sigma^j$ be normalized families of complex $d\times d$ matrices indexed by $a<r,x<s$: this means $d,r,s>0$, every block is positive semidefinite, the sum over $a$ is independent of $x$, and the mass is one, where $T_A=s^{-1}\sum_{x,a}A_{a|x}$ and $M_A=\operatorname{Re}\operatorname{tr}T_A$; the underlying assemblage condition also requires $0\le M_A\le1$. Define the mixtures blockwise by $\bar\rho_{a|x}=\sum_{j<n}p_j\rho^j_{a|x}$ and $\bar\sigma_{a|x}=\sum_{j<n}p_j\sigma^j_{a|x}$. The entropy $\mathcal E(A,B)$ used here is defined as follows. For square matrices $X,Y$ of the same finite size, let $\mathcal D(X\Vert Y)=\infty$ unless $X,Y$ are both positive semidefinite with trace one and $\ker Y\subseteq\ker X$; when those conditions hold, let it be the nonnegative part of $\operatorname{Re}\operatorname{tr}(f(X)-X\log Y)$, where $f(t)=t\log t$, matrix functions use functional calculus, and $\log0=0$. Write $\mathcal D_{\mathrm{cl}}(v\Vert w)=\mathcal D(\operatorname{diag}v\Vert\operatorname{diag}w)$. For every entropy strategy consisting of a natural number $h$, matrices $L_g\in\mathbb C^{d\times d}$ with $\sum_{g<h}L_g^\dagger L_g=I$, and nonnegative numbers $u_{g,x}$ with $\sum_{x<s}u_{g,x}=1$, put $v_g=\operatorname{Re}\operatorname{tr}(L_gT_AL_g^\dagger)$, $w_g=\operatorname{Re}\operatorname{tr}(L_gT_BL_g^\dagger)$, and let $F^A_{g,x},F^B_{g,x}$ be block diagonal matrices with respective outcome blocks $L_gA_{a|x}L_g^\dagger$ and $L_gB_{a|x}L_g^\dagger$. Put $C_{g,x}=v_g\mathcal D(F^A_{g,x}/v_g\Vert F^B_{g,x}/w_g)$ when $v_g>0,w_g>0$, put $C_{g,x}=\infty$ when $v_g>0,w_g\le0$, and put $C_{g,x}=0$ when $v_g\le0$. Then $\mathcal E(A,B)$ is the supremum, over all such finite strategies, of $\mathcal D_{\mathrm{cl}}(v\Vert w)+\sum_{g<h,x<s}u_{g,x}C_{g,x}$. The declaration asserts

The inequality is in the extended nonnegative reals, real nonnegative weights are embedded into that space, and $0\cdot\infty=0$; the entropies need not be finite. The hypotheses exclude both an empty mixture and zero mixture weights.