Refined Severi degrees on classical toric del Pezzo surfaces

Let $S$ be a classical toric del Pezzo surface and let $L$ be a line bundle. Assume that the loci in $|L|$ of non-reduced curves and of curves having a $(-1)$-curve as a component both have codimension greater than $\delta$. Let $N^{(S,L),\delta}(y)$ be the refined Severi degree and $\widetilde N^{(S,L),\delta}(y)$ the refined universal invariant. Refined toric del Pezzo conjecture.

$$N^{(S,L),\delta}(y)=\widetilde N^{(S,L),\delta}(y).$$

The corresponding unrefined assertion is cited as proved, while the refined version is presented as an expectation.