Conjecture on differentiability of the equilibrium measure

Let $\tilde{\gamma}_j$ be neighborhoods of the connected components $\gamma_j$ of the support of the equilibrium measure, and set $\tilde{\Gamma}=\coprod_{j=1}^r\tilde{\gamma}_j$. Let $h:\tilde{\Gamma}\rightarrow\mathbb{R}$ be an admissible function, let $\epsilon\in(\mathbb{R}_+^\times)^r$ satisfy $\sum_{j=1}^r\epsilon_j=1$, and let $\mu_{\mathrm{eq}}[V,\epsilon]$ denote the unique minimizer of the strictly convex energy functional among probability measures with masses $\epsilon_j$ on $\tilde{\gamma}_j$. Differentiability conjecture. If Hypothesis~ holds, then for generic $\mathcal{V}$ and generic $\epsilon$, there exists a linear map $\mu'_{\mathrm{eq}}[V,\epsilon]$ on triples $(h,\delta,f)$, where $\delta\in\mathbb{R}^r$ satisfies $\sum_{j=1}^r\delta_j=0$ and $f:\tilde{\Gamma}\rightarrow\mathbb{R}$ is bounded and continuous, such that

$$\mu'_{\mathrm{eq}}[V,\epsilon]\cdot(h,\delta,f)=\lim_{t\rightarrow0}\frac{1}{t}\int_{\tilde{\Gamma}}f(x)\,\mathrm{d}\mu_{\mathrm{eq}}[V+th,\epsilon+t\delta](x).$$

This conjecture formalizes the expected $\mathcal{C}^1$ dependence of the equilibrium measure on the potential and filling fractions. The precise functional-analytic proof is not supplied, and the statement depends on the regularity hypothesis referenced in the source.