Kolesnikov–Milman local $L_p$-Minkowski conjecture

Let $n\geq2$, let $p\in[0,1)$, and let $K$ be an origin-symmetric smooth strictly convex body with support function $h_K$. Write $H_K=\nabla^2h_K+h_K\operatorname{Id}$ and let $(H_K^{ij})$ be its inverse. Let $C^2_e(S^n)$ denote the even twice continuously differentiable functions on $S^n$.

Kolesnikov–Milman local $L_p$-Minkowski conjecture. For every $z\in C^2_e(S^n)$ satisfying

$$\int_{S^n}zh_K\,dS_K=0,$$

one has

$$\int_{S^n}H_K^{ij}z_iz_jh_K^2\,dS_K\geq(n+1-p)\int_{S^n}z^2h_K\,dS_K.$$

At $p=0$, this is the local log-Minkowski conjecture. It is a spectral and variational strengthening of the local uniqueness problem for the even $L_p$-Minkowski problem, and the source does not report a resolution for the full range $p\in[0,1)$.