Banks–Martin conjecture for Erdős sums

For each positive integer $k$, let $\mathcal{P}_k$ be the set of natural numbers with exactly $k$ prime factors counted with multiplicity, and let $\mathcal{F}$ denote the Erdős sum in the integers.

Banks–Martin conjecture. The sequence of sums is strictly decreasing:

$$\mathcal{F}(\mathcal{P}_1)>\mathcal{F}(\mathcal{P}_2)>\cdots>\mathcal{F}(\mathcal{P}_k)>\mathcal{F}(\mathcal{P}_{k+1})>\cdots.$$

The conjecture is a refinement of the primitive set problem, comparing sets with a fixed number of prime factors. It was recently disproved by Lichtman.