Measure classification from monotonicity of boundary content under Minkowski addition

Let $n\geq 2$, let $\mu$ be a Radon measure on $\mathbb R^n$, and write $\mu^+(\partial K)$ for the boundary-content functional used in the source. Assume that for every convex body $K$ and every compact, convex set $L$ containing the origin,

$$\mu^+(\partial(K+L))\geq\mu^+(\partial K).$$

Measure-classification conjecture. Then $\mu$ is a constant multiple of Lebesgue measure.

This proposed extension of the one-dimensional result would recover the classification of Lebesgue measure in higher dimensions. The supplied text presents it as an unresolved question and gives no resolution.