Canonical singularities conjecture for Kontsevich moduli spaces

Let $X\subset \mathbb{P}^n$ be a general hypersurface of degree $d$, and let $\overline{\mathcal{M}}_{0,0}(X,e)$ be the Kontsevich moduli space of degree-$e$ stable maps to $X$. Canonical singularities conjecture. If $e\geq 3$ and $d+e\leq n$, then $\overline{\mathcal{M}}_{0,0}(X,e)$ is a normal, $\mathbb{Q}$-Gorenstein variety with only canonical singularities. If $e=2$, $d+3\leq n$, then $\overline{\mathcal{M}}_{0,0}(X,2)$ has the same properties. This is the corresponding assertion for a general hypersurface, strengthening the established stack-theoretic results discussed earlier; the supplied text gives no resolution status.