Asymptotic F-nefness conjecture for differences of Virasoro conformal-block divisors

Fix $r>0$. For integers $k$, $n$, and labels $a_1,\ldots,a_n$, consider the difference of conformal-block divisors on $\overline{\mathcal{M}}_{0,n}$ associated with the Virasoro vertex operator algebras $\text{Vir}_{2k+3,2}$ and $\text{Vir}_{2k+1,2}$. Asymptotic F-nefness conjecture. For every $r>0$, there exists $N>0$ such that, for every $n\geq N$, the divisor

$$\mathbb{D}_{0,n}\left(\text{Vir}_{2k+3,2},\bigotimes_{i=1}^{n}W_{a_i}\right)-\mathbb{D}_{0,n}\left(\text{Vir}_{2k+1,2},\bigotimes_{i=1}^{n}W_{a_i}\right)$$

is F-nef whenever

$$2\leq a_1,\ldots,a_n\leq 2k,\qquad \sum_{i=1}^{n}(a_i-1)\text{ is odd},\qquad l(a_1,\ldots,a_n)-r\leq k.$$

Here $l(a_1,\ldots,a_n)$ is the level quantity used in the source, and F-nef means nonnegative intersection with every F-curve. The conjecture is motivated by computational experiments suggesting stronger nefness properties for differences of Virasoro conformal-block divisors; no resolution is given.