Expected dimension conjecture for maps to general hypersurfaces in Grassmannians

Let $C$ be a smooth curve of genus $g$, let $G(r,n)$ be a Grassmannian, and let $X_{\ell}$ be a general hypersurface of degree $\ell$ in $G(r,n)$ with $\ell<n$. Denote by $\mathrm{Mor}_d(C,X_{\ell})$ the space of maps of degree $d$ from $C$ to $X_{\ell}$. Expected dimension conjecture. There exists a threshold degree $d_0(g,r,n)$ such that $\mathrm{Mor}_d(C,X_{\ell})$ has expected dimension for all $d>d_0$. This question concerns when the virtual map counts discussed in the paper become enumerative for sufficiently large degree; the source does not provide evidence of a resolution.