Farkas's codimension conjecture for syzygy strata of curves

Let $\mathcal{M}_g^0$ be the open moduli space under consideration, and let $\mathcal{Z}_{g,i}$ be the locus of curves $C$ for which there exists a pencil $A\in W^1_k(C)$ such that the residual pair $(C,K_C\otimes A^{\vee})$ fails property $(N_i)$. Farkas's codimension conjecture for syzygy strata. For an even genus $g\geq 6i+10$, the stratum $\mathcal{Z}_{g,i}$ is a proper subvariety of $\mathcal{M}_g^0$. In particular, when $g=6i+10$, the stratum $\mathcal{Z}_{g,i}$ is a divisor on $\mathcal{M}_g^0$. The claim predicts the expected properness of these residual-pencil syzygy strata and, in the balanced case, their divisorial nature.