The large-genus Brill–Noether containment threshold conjecture for $r<s$

Let $\mathcal{M}^{r}_{g,d}$ and $\mathcal{M}^{s}_{g,e}$ denote Brill–Noether loci of curves of genus $g$ carrying respectively a $g^r_d$ and a $g^s_e$. Assume $g$ and $d$ are sufficiently large, $d,e\leq g-1$, and $2\leq r<s$. Define

$$T=d-2r+s+\frac{g-d+r+1}{2}+\frac{(s-2)(r-1)-1}{s-1}.$$

Large-genus containment threshold conjecture. If $e<T$, then

$$\mathcal{M}^{r}_{g,d}\nsubseteq\mathcal{M}^{s}_{g,e},$$

whereas if $e\geq T$, then

$$\mathcal{M}^{r}_{g,d}\subset\mathcal{M}^{s}_{g,e}.$$

This conjecture predicts a sharp threshold between non-containment and containment when the target rank exceeds the source rank. It is motivated by K3-surface admissible-assignment bounds; the source presents it as an expectation rather than a proved result.