Dimension conjecture for Kontsevich spaces on general Fano hypersurfaces

Let $X$ be a general hypersurface of degree $d$ in $\mathbb{P}^n$. Let $\overline{\mathcal{M}}_{0,0}(X,e)$ denote the Kontsevich space of degree $e$ stable rational maps to $X$. Dimension conjecture. The dimension of $\overline{\mathcal{M}}_{0,0}(X,e)$ is

$$\max\{0,e(n-d+1)+n-4\},$$

the minimum possible. This is an open question, conjectured by Coskun, Harris and Starr in the special case $n\geq d+1$, and remains open for large ranges of $d$, $n$ and $e$.