Huneke's finiteness conjecture for Bass numbers of local cohomology modules

Let $(R,\mathfrak{m})$ be a regular local ring, let $\mathfrak{a}$ be an ideal of $R$, and let $\mathfrak{p}\in\operatorname{Spec}(R)$. For $i,j\in\mathbb{N}_0$, define the Bass numbers

$$\mu^i(\mathfrak{p},H^j_{\mathfrak{a}}(R)):=\operatorname{dim}_{\kappa(\mathfrak{p})}\left(\operatorname{Ext}^i_{R_{\mathfrak{p}}}(\kappa(\mathfrak{p}),H^j_{\mathfrak{a}}(R)_{\mathfrak{p}})\right).$$

Huneke's conjecture. The Bass numbers $\mu^i(\mathfrak{p},H^j_{\mathfrak{a}}(R))$ are finite for all $i,j\in\mathbb{N}_0$ and every $\mathfrak{p}\in\operatorname{Spec}(R)$. This conjecture asserts finiteness of all Bass numbers of local cohomology modules over regular local rings; it is a central finiteness problem in local cohomology, with known results in various cases but not resolved in the stated generality.