Cartwright–Strumfels local-cohomology invariance conjecture

Let $A$ be a polynomial ring, let $I\subseteq A$ be a $\mathbb{Z}^m$-graded Cartwright–Strumfels ideal, let $\operatorname{gin}(I)$ be its $\mathbb{Z}^m$-graded generic initial ideal, and let $\mathfrak m$ be the relevant homogeneous maximal ideal. Cartwright–Strumfels invariance conjecture. For every $r\in\mathbb N$ and every $a\in\mathbb Z^m$,

$$\dim_K H_{\mathfrak m}^r(A/I)_a=\dim_K H_{\mathfrak m}^r(A/\operatorname{gin}(I))_a.$$

The source attributes the assertion to work comparing Cartwright–Strumfels ideals with their generic initial ideals, but does not state whether it is open or resolved.