Convergence under eventual convexity and a unique single-support minimizer

Let $V$ be a $C^2$ potential function, and let $K_1,K_2,r>0$ satisfy

$$V”(x)\geq K_1\quad\text{for all }|x|\geq r,$$

and

$$V”(x)\geq-K_2\quad\text{for all }|x|\leq r.$$

Suppose that $\Sigma_V$ has a unique minimizer $\mu_V$ whose support is a single compact set. Convergence conjecture. Then $\mu_t$ converges to $\mu_V$ with respect to the $W_2$-Wasserstein distance and in the weak convergence topology on $\mathscr{P}(\mathbb{R})$. This would extend the known convergence results for convex potentials to a class that is only eventually convex; the required global convergence remains open.