The general secant non-containment conjecture for Brill–Noether loci

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 $g$ is sufficiently large, $d,e\leq g-1$, and $r\geq s+1\geq 3$.

General secant non-containment conjecture. If

$$e<d-2r+2+s+\left\lfloor\frac{r-1}{2}\right\rfloor,$$

then

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

The source proves containments in part of the complementary secant-expected range. Outside that range, the asserted non-containments are posed as a conjecture and remain open.