Farkas's pure-resolution conjecture for special curves

Let $p\ge 0$ and $s\ge 2$ be integers, and set

$$r=(s+1)(p+2)-2,\qquad g=(r+1)s,\qquad d=r(s+1).$$

Let $C\subset\mathbb P^r$ be a general smooth normal curve of genus $g$, degree $d$, and speciality $h^1(\mathcal O_C(1))=s$.

Farkas's conjecture. The curve $C$ has a pure resolution, equivalently it satisfies property $N_p$.

If true, the associated non-$N_p$ locus gives divisors that yield counterexamples to the slope conjecture. The source reports computational verification in small cases, but the supplied status is unknown.