Primary full-system conjecture for finite-rank submonoids

For $d\geq 3$, let $\mathcal{C}_d$ be the class of rank-$d$ submonoids of finite-rank free commutative monoids. A monoid has a full system of sets of lengths when its system of sets of lengths is $\mathbb{P}_{\mathrm{fin}}$, the collection of all finite subsets of positive integers. A monoid is primary when it has no nonempty proper divisor-closed submonoid.

Primary full-system conjecture. For every dimension $d\geq 3$, there exists a primary monoid in $\mathcal{C}_d$ having full system of sets of lengths.

The preceding construction gives, for every $d\geq 3$, a rank-$d$ monoid with full system of sets of lengths, but that monoid is not primary. The conjecture asks whether primarity can be achieved while retaining the same extremal factorization property.