The secant non-containment conjecture for Brill–Noether loci of rank two

Let $\mathcal{M}^{r}_{g,d}$ denote the Brill–Noether locus of curves of genus $g$ carrying a $g^r_d$. Assume $g$ is sufficiently large, $r\geq 3$, and $d,e\leq g-1$.

Secant non-containment conjecture. If

$$e<d-2r+2+\left\lfloor\frac{r+3}{2}\right\rfloor,$$

then

$$\mathcal{M}^{r}_{g,d}\nsubseteq\mathcal{M}^{2}_{g,e}.$$

The preceding theorem establishes containments in the complementary secant-expected range. Computations on K3 surfaces support this conjecture, and Lelli–Chiesa verifies it under additional restrictions for a subrange; the full statement remains open.