Conjecture on non-Gaussian minimizers in the variational problems

Let $p\in(2,\infty)$. For $w\in(-1,1)\setminus\{0\}$, consider the variational problem

$$\inf_{\nu\in\mathcal{P}(\mathbb{R})}\left\{\mathbb{I}^{\mathsf{an}}_{\infty,\mu_p}(w)+H(\nu\mid\mu_2)+\frac{1}{2}(1-m_2(\nu))\right\},$$

with the minimizer understood as in. For $\gamma\in\mathcal{T}_p$ and $w\in D_{\mathbb{I}^{\mathsf{an}}_{\gamma}}\setminus\{0\}$, consider analogously the variational problem in.

Non-Gaussian minimizer conjecture. For every $p\in(2,\infty)$ and $w\in(-1,1)\setminus\{0\}$, the minimizer in is not $\mu_2$. Similarly, for every $\gamma\in\mathcal{T}_p$ and $w\in D_{\mathbb{I}^{\mathsf{an}}_{\gamma}}\setminus\{0\}$, the minimizer in is not $\mu_2$. Consequently, except at $w=0$, the annealed rate function lies strictly below the quenched rate function.

This conjecture would extend the preceding proposition from the Gaussian setting to all $p\in(2,\infty)$ and to products of measures other than $\gamma=\mu_\infty$. Its resolution would clarify when the annealed and quenched rate functions differ strictly away from the origin.