Eventual linearity of associated-prime counts for Inc-invariant monomial chains

Let ${\mathcal I}=(I_n)_{n\ge 1}$ be an $\operatorname{Inc}$-invariant chain of monomial ideals. Associated-prime count conjecture. The number of associated primes of $I_n$ is eventually a linear function of $n$. This expectation arises from computational experiments and is presented as an open problem related to the regularity conjecture; no proof or disproof is given in the source.