Finiteness conjecture for cylinder-width triples in the hyperelliptic stratum

Let $C$ be an algebraically primitive Teichmüller curve in the hyperelliptic stratum ${\Omega\mathcal M}_{3}(4)^{{\rm hyp}}$, and let $(r_1:r_2:r_3)$ be the projectivized triple of widths of its cylinders. Finiteness conjecture. There are only a finite number of possibilities for the projectivized triples $(r_1:r_2:r_3)$ of widths of cylinders of algebraically primitive Teichmüller curves in ${\Omega\mathcal M}_{3}(4)^{{\rm hyp}}$. This conjecture supplies numerical and theoretical evidence toward the genus-three finiteness conjecture in the hyperelliptic stratum, and together with the cited proposition is stated to imply that case of the broader conjecture; it remains open in the source.