The lower bounded k-th mean curvature characterization of balls

Let $K\subseteq\mathbf{R}^{n+1}$ be a convex body. For $k=1,\ldots,n$, let $H_k(K,\cdot)$ denote its pointwise $k$-th mean curvature, and write $\mathscr{H}^{n}$ and $\mathscr{L}^{n+1}$ for Hausdorff and Lebesgue measure, respectively. Ball characterization conjecture. If

$$H_k(K,x)\geq\bigg(\frac{\mathscr{H}^{n}(\partial K)}{(n+1)\mathscr{L}^{n+1}(K)}\bigg)^k{n\choose k}$$

for $\mathscr{H}^{n}$-almost every $x\in\partial K$, then $K$ is a ball. This conjecture proposes a characterization of the ball using only a pointwise lower bound on the $k$-th mean curvature; the source gives no resolution, so its status remains open.