Bhatt–Hochster–Ma conjecture on lim Cohen–Macaulay sequences

Let $R$ be a Noetherian complete local domain whose residue field is algebraically closed. A lim Cohen–Macaulay sequence is the structure introduced by Bhatt, Hochster, and Ma as a substitute for small or maximal Cohen–Macaulay modules.

Bhatt–Hochster–Ma conjecture. The ring $R$ has a lim Cohen–Macaulay sequence.

If true, this conjecture would imply Serre's positivity conjecture on multiplicities in general. The source presents it as an open conjecture and refers to Hochster for a precise statement.