Lower semicontinuity conjecture for convex integral functionals of matricial measures

Let $\mathfrak{H}_{+,m}$ denote the cone of positive semidefinite Hermitian $m\times m$ matrices, let $\mathbb T^d$ be the $d$-dimensional torus, and let $\mathrm{d}M$ be a nonnegative matricial measure on $\mathbb T^d$. Write its Lebesgue decomposition as $\mathrm{d}M=\Phi(\boldsymbol{\theta})\,\mathrm{d}\mu(\boldsymbol{\theta})+\mathrm{d}M_{\mathrm{s}}$, where $\Phi$ is the absolutely continuous part and $\mu$ is Lebesgue measure. Let $f:\mathfrak{H}_{+,m}\times\mathbb T^d\to\mathbb R$ be such that $f(\,\cdot\,,\boldsymbol{\theta})$ is convex in its first argument for every fixed $\boldsymbol{\theta}\in\mathbb T^d$. Lower semicontinuity conjecture. The functional

$$g:\mathrm{d}M\longmapsto\int_{\mathbb T^d}f(\Phi(\boldsymbol{\theta}),\boldsymbol{\theta})\,\mathrm{d}\mu(\boldsymbol{\theta})$$

is lower semicontinuous with respect to the weak* topology. The conjecture is intended to provide a mathematical basis for the chosen divergence index and to ensure existence of a solution to the associated primal problem when the prior is arbitrary; the source discussion does not provide a proof, and its resolution remains open.