Oeding's Cohen–Macaulayness conjecture for secant varieties of Segre products

Let $\mathbb{P}V_1,\ldots,\mathbb{P}V_k$ be projective spaces, let $X=\operatorname{Seg}(\mathbb{P}V_1\times\cdots\times\mathbb{P}V_k)$ be their Segre product, and let $\widehat{\sigma}_r$ be the affine cone over its $r$-th secant variety $\sigma_r(X)$. A projective variety is arithmetically Cohen–Macaulay when its affine cone is Cohen–Macaulay.

Oeding's conjecture. All secant varieties of Segre products of projective spaces are arithmetically Cohen–Macaulay; equivalently, $\widehat{\sigma}_r$ is Cohen–Macaulay for every $r$.

Cohen–Macaulayness is known in many special cases, but the statement is presented in the source as a broad conjecture extending these partial results to all secant varieties of Segre products.