Conca–De Negri–Gorla local cohomology conjecture for Cartwright–Sturmfels ideals

Let $A=\mathbb{K}[x_1,\dots,x_n]$ be a polynomial ring over a field $\mathbb{K}$, let $I\subseteq A$ be a $\mathbb{Z}^m$-graded Cartwright–Sturmfels ideal, and let $\operatorname{gin}(I)$ be its $\mathbb{Z}^m$-graded generic initial ideal, where $m\leq n$. For $r\in\mathbb{N}$ and $a\in\mathbb{Z}^m$, write $H^r_{\mathfrak m}(A/I)_a$ for the multigraded component of the local cohomology module supported at the homogeneous maximal ideal $\mathfrak m$. Conca–De Negri–Gorla's local cohomology conjecture. For every $r\in\mathbb{N}$ and every $a\in\mathbb{Z}^m$, one has

$$\dim_\mathbb{K} H^r_{\mathfrak m}(A/I)_a=\dim_\mathbb{K} H^r_{\mathfrak m}(A/\operatorname{gin}(I))_a.$$

If true, this would extend the comparison between Cartwright–Sturmfels ideals and their generic initial ideals to all multigraded local cohomology components. The supplied source does not state whether the conjecture has been resolved.