Gap-free floor conjecture for semiprime subset sums

Let $N$ be a positive integer, let $P_N$ be the product of the primes at most $N$, and let $L^2(N)$ denote the set of subset sums of the corresponding semiprime unit-fraction contributions. Define

$$\gamma_N=\inf\{\gamma:[\gamma P_N,\,\sigma^2(N)-\gamma P_N]\cap\mathbb{Z}\subseteq L^2(N)\}.$$

Gap-free floor conjecture.

$$\gamma_N\to0\quad\text{as }N\to\infty.$$

This conjecture asserts that the semiprime subset sums become gap-free throughout the bulk of the interval $[0,\sigma^2(N)]$, apart from a vanishing proportion near its two ends. It identifies the remaining open core of the $\omega=2$ case of the Erdős–Graham problem; the supplied source does not state a resolution.