Interior boundary-cone conjecture for Virasoro conformal-block divisors

Let $V=\text{Vir}_{2,2k+1}$, and let $\{W_{a_i}\}_{i=1}^n$ be a set of nontrivial simple $V$-modules, with $a_i\neq 1$. Let $\mathbb{D}_{0,n}\left(V,\bigotimes_{i=1}^n W_{a_i}\right)$ denote the associated conformal-block divisor on $\overline{\mathcal{M}}_{0,n}$. Interior boundary-cone conjecture. Either

$$\mathbb{D}_{0,n}\left(V,\bigotimes_{i=1}^n W_{a_i}\right)=0,$$

or its negative lies in the interior of the cone generated by the boundary divisors:

$$-\mathbb{D}_{0,n}\left(V,\bigotimes_{i=1}^n W_{a_i}\right)\text{ lies in the interior of the cone generated by the boundary divisors}.$$

The conjecture is proposed as a criterion used to prove nefness for specific values of $k$; the source gives no general resolution.