Banks–Martin conjecture for odd-prime-supported primitive sets

Let $\Omega(n)$ be the number of prime factors of $n$, counted with multiplicity, and let $\mathbb{N}_k=\{n:\Omega(n)=k\}$. For a set of odd primes $\mathcal Q$, write $A(\mathcal Q)$ for the members of $A$ composed only of primes in $\mathcal Q$, and similarly let $\mathbb{N}_k(\mathcal Q)$ denote the members of $\mathbb{N}_k$ composed only of primes in $\mathcal Q$. Banks–Martin conjecture. If $k\ge1$ and $A$ is a primitive set such that $\Omega(n)\ge k$ for every $n\in A$, then for every set of odd primes $\mathcal Q$,

$$f(A(\mathcal Q))\le f\big(\mathbb{N}_k(\mathcal Q)\big).$$

This is presented as a broad generalization of the Erdős primitive set conjecture and as a related open question. The source gives no resolution of this formulation.