Polynomial growth conjecture for the length-ratio invariant

Let $S_{g,m}$ be a surface and let $\rho\in\operatorname{Pos}_3^h(S_{g,m})$ have canonical area at most $t$. For every non-trivial, non-peripheral $\gamma\in\pi_1(S_{g,m})$, let $\ell_1^\rho(\gamma)$ and $\ell_2^\rho(\gamma)$ be the associated length invariants. There is a constant $L(t)$, independent of $\rho$, such that

$$\frac{\ell_2^\rho(\gamma)}{\ell_1^\rho(\gamma)}\leq L(t).$$

Polynomial growth conjecture. The constant $L(t)$ above can be a polynomial of $t$.

This conjecture asks for a quantitative polynomial version of the established uniform boundedness of the ratio $\ell_2^\rho(\gamma)/\ell_1^\rho(\gamma)$ for bounded-area positive convex structures. The supplied text does not indicate whether it has been resolved.