Density-realization conjecture for weak saturation limits

Let $\delta\geq2$ be fixed, and let $w\in\mathbb Q\cap[\delta/2,\delta-1]$. Density-realization conjecture. There exists a graph $F$ such that

$$\delta_F=\delta\qquad\text{and}\qquad w_F=w.$$

This would extend the dense-range realization theorem in the paper from the cases established there; the source presents it as another problem for future research, so the general assertion remains open.