Conjecture on the perfection and linear syzygies of the principal-curve ideal

Let the subhomaloidal type be $(s;(s-1)/2^3,\mu_4,\ldots,\mu_r)$ with corresponding proper homaloidal type, and let $f,g,q$ be the three principal curves described in the paper. Set $R=k[x,y,z]$ and consider the ideal $(f,g,q)\subset R$.

Principal-curve ideal conjecture. The ideal $(f,g,q)$ is perfect, that is, unmixed. Moreover, it admits a linear syzygy if and only if either $s=3$ or $s=5$, or else $s>5$ and $\mu_4=(s-1)/2$.

A resolution of this question would clarify when the six forms built from $f,g,q$ can be treated uniformly in the birational and homological arguments. The source presents the assertion as conjectural and gives no indication of a resolution.