Berkovich–Uncu's corrected nonnegativity conjecture for $G_{L,2}(q)$

For a positive integer $L$, let $G_{L,2}(q)$ be the difference of the generating series for partitions with smallest part $2$ and largest-minus-smallest part at most $L$, and for partitions with smallest part at least $3$ and largest-minus-smallest part at most $L$. Write $S\succeq 0$ when every coefficient of the series $S$ is nonnegative. Berkovich and Uncu's conjecture.

$$L=3\implies G_{L,2}(q)+q^3+q^9+q^{15}\succeq 0;$$ $$L=4\implies G_{L,2}(q)+q^3+q^9\succeq 0;$$ $$L\geq 5\implies G_{L,2}(q)+q^3\succeq 0.$$

The source explicitly says that it proves this conjecture in the paper, so the claim is solved.