The obstruction conjecture for C^2 Lusin approximation of convex functions

Let $n\geq 2$. A convex function is a function whose restriction to every line segment in its domain is convex. Write $B^n(0,1)$ for the open unit ball in $\mathbb R^n$, and let $|\cdot|$ denote Lebesgue measure. Obstruction conjecture. There is a convex function

$$f:B^n(0,1)\to\mathbb R$$

and a number $m>0$ such that, for every convex function $g\in C^2(B^n(0,1))$,

$$|\{f\neq g\}|\geq m.$$

This conjectures that in dimensions at least two, some convex functions cannot be approximated in the Lusin sense by convex $C^2$ functions, complementing the one-dimensional approximation theorem and the known result for locally strongly convex functions. The general question remains open.