Ash–Catoiu conjecture on directional and relative Lipschitz continuity

Let $f:\mathbb{R}^2\to\funkmathbb{R}$, and let

$$E=\bigl\{\boldsymbol{x}\in\mathbb{R}^2: f\text{ is Lipschitz at }\boldsymbol{x}\text{ in every direction}\bigr\}.$$

Define

$$F=\bigl\{\boldsymbol{x}\in E: f\text{ is not Lipschitz at }\boldsymbol{x}\text{ relative to }E\bigr\}.$$

Ash–Catoiu conjecture. The set $F$ is a null set.

This conjecture concerns the relationship between directional Lipschitz continuity and Lipschitz continuity relative to the set of points where directional Lipschitz continuity holds. The source paper states that it proves a stronger version of this conjecture, so the claim is resolved by the cited result.