Delta-conjecture for the nef cone of blown-up planes

Let $X$ be the blowup of $\mathbb{P}^2$ at $n$ sufficiently general points. Let $\mathcal{Q}_n$ be the nonnegative cone and let $\Delta_n^{\preccurlyeq}$ be the cone defined in the source. Delta-conjecture. If $n\ge10$, then

$$\partial\mathcal{Q}_n\cap\Delta_n^{\preccurlyeq}\subset\operatorname{Nef}(X).$$

This conjecture controls part of the boundary of the nef cone and implies a corresponding equality involving the Mori and nef cones in the indicated region. It remains open in the source.