The converse characterization of convex Lipschitzian functions

Let $X$ be the domain of a Lipschitzian function $F:X\to\mathbb{R}$, and let $\nabla F(\boldsymbol{x})$ denote the set of subgradients of $F$ at $\boldsymbol{x}$. For $\boldsymbol{x},\boldsymbol{y}\in X$ and $0\leq t\leq 1$, assume that $\nabla F(\boldsymbol{x})\cap\nabla F(\boldsymbol{y})\neq\varnothing$. Convexity characterization. The function $F$ is convex if and only if

$$\nabla F(\boldsymbol{x})\cap\nabla F(\boldsymbol{y})=\nabla F(t\boldsymbol{x}+(1-t)\boldsymbol{y}).$$

This is proposed as the converse of the preceding convexity property for subgradient sets; the supplied text does not state whether the characterization has been proved or remains open.