Sharp Michael–Simon inequality conjecture for higher mean curvatures

Let $M$ be a compact hypersurface in $\mathbb{R}^{n+1}$, possibly with boundary $\partial M$, and let $f$ be a positive smooth function on $M$. For $1\leq k\leq n-1$, write $\sigma_k=\sigma_k(\kappa)$ for the $k$th mean curvature, where $\kappa$ denotes the principal curvatures of $M$. Sharp Michael–Simon inequality conjecture. One has

$$\int_M \sqrt{\sigma_k^2f^2+\sigma_{k-1}^2|\nabla^M f|^2}+\int_{\partial M}\sigma_{k-1}f \geq n|B^n|^{\frac{1}{n-k+1}}\left(\int_M\sigma_{k-1}f^{\frac{n-k+1}{n-k}}\right)^{\frac{n-k}{n-k+1}}.$$

Equality holds if and only if $M$ is a sphere and $f$ is constant. This is proposed as the sharp Michael–Simon-type inequality for the $k$th mean curvature in Euclidean space; the source motivates it by known inequalities for higher mean curvatures but provides no resolution of the conjecture.