Compactness of the minimal elements of quasicomplemented posets

Let $Q$ be a poset, and write $Min(Q)$ for its set of minimal elements. Suppose that $Q$ is quasicomplemented and belongs to the class $\mathbb{P}^{\ell}_{MFP}$, and that it satisfies the condition $a.c.$

Compactness conjecture. Is $Min(Q)$ compact?

The question is motivated by the preceding example of a quasicomplemented poset in $\mathbb{P}^{\ell}_{MFP}$ without $a.c.$ for which $Min(Q)$ is not compact; the conjecture asks whether the additional $a.c.$ hypothesis guarantees compactness.