Equivalence of metric regularity and strong metric regularity of the subdifferential

Let $f\colon\mathbb{R}^n\to\overline{\mathbb{R}}$ be both prox-regular and subdifferentially continuous at $\bar{x}$ for $\bar{x}^*=0$, where $\bar{x}$ is a local minimizer of $f$. The subdifferential mapping $\partial f$ is metrically regular at $(\bar{x},\bar{x}^*)$ if and only if it is strongly metrically regular there.

Equivalence of metric regularity and strong metric regularity. Under these hypotheses, metric regularity and strong metric regularity of the subdifferential are the same notions.

For convex functions, for twice continuously differentiable functions, and for functions of the form $f(x)=f_0(x)+\delta_\Omega(x)$ with $f_0$ twice continuously differentiable and $\Omega$ polyhedral convex, the equivalence is known; the conjecture concerns all prox-regular and subdifferentially continuous functions on finite-dimensional spaces.