Hellus–Lyubeznik–Yildirim's conjecture on local cohomology modules

Let $(R,\mathfrak{m},\mathbb{K})$ be a commutative Noetherian local ring with residue field $\mathbb{K}=R/\mathfrak{m}$, let $I\subset R$ be an ideal, and let $H_I^j(R)$ denote the $j$-th local cohomology module supported on $I$. For an $R$-module $M$, write $M^{\vee}=\operatorname{Hom}_R(M,E)$ for its Matlis dual, where $E$ is an injective hull of the residue field. Let $\operatorname{Ass}_R(M)$ denote the set of associated prime ideals of $M$. Hellus–Lyubeznik–Yildirim's conjecture. Assume, in addition, that $R$ is regular, and let $j\geq 0$. Then, if $H_I^j(R)\neq 0$, then $(0)\in\operatorname{Ass}_R\left(H_I^j(R)^{\vee}\right)$. This conjecture predicts that every nonzero local cohomology module over a regular local ring has Matlis dual whose associated-prime set contains the zero ideal. The paper gives partial positive answers under hypotheses such as $\operatorname{depth}(R/I)=1$, certain cases with depth two, squarefree monomial ideals in formal power series rings over a field, and formal power series rings over discrete valuation rings of mixed characteristic; the general conjecture remains open.