Farkas's codimension conjecture for Brill–Noether strata of spin curves

Let $\mathcal{S}_g$ be the moduli space of theta characteristics, and for an integer $r\geq -1$ define

$$\mathcal{S}_g^r:=\bigl\{[C,\theta]\in\mathcal{S}_g:h^0(C,\theta)\geq r+1,\ h^0(C,\theta)\equiv r+1\pmod 2\bigr\}.$$

Farkas's codimension conjecture. For $r\geq 1$ and $g\geq {r+2\choose 2}$, there exists a component of $\mathcal{S}_g^r$ having codimension ${r+1\choose 2}$ inside $\mathcal{S}_g$. The prediction concerns the expected codimension and regularity of Brill–Noether strata of theta characteristics; the surrounding discussion explains that the bound is not universally realized, while this precise assertion is proposed when $r$ is relatively small compared with $g$.