Huneke–Lyubeznik finiteness conjecture for associated primes of local cohomology

Let $R$ be a regular ring and let $I \subseteq R$ be an ideal. For an $R$-module $M$, write $\operatorname{Ass}_R(M)$ for its set of associated prime ideals.

Huneke–Lyubeznik conjecture. For every $i \ge 0$, the local cohomology module $H^i_I(R)$ has finitely many associated prime ideals.

This conjecture has been open for three decades. It is known for regular rings in equal characteristic $p>0$, and in equal characteristic zero when $R$ is local or affine; the paper proves substantial new cases for excellent regular rings of finite Krull dimension.