Polynomial growth conjecture for the diagonal-edge invariant

Let $S_{g,m}$ be a surface and let $\rho\from S_{g,m}\to\mathbb{RP}^2$ be a bounded positive convex structure whose canonical area is at most $t$. For an ideal quadrilateral embedded in a pair of pants, let $\overrightarrow{\mathbf{e}}$ be its oriented diagonal ideal edge in an ideal triangulation, and let $D_1(\overrightarrow{\mathbf{e}})(\rho)$ and $D_2(\overrightarrow{\mathbf{e}})(\rho)$ be the associated invariants. A uniform bound $D(t)$ satisfies

$$\left|\log D_1(\overrightarrow{\mathbf{e}})(\rho)-\log D_2(\overrightarrow{\mathbf{e}})(\rho)\right|\leq D(t).$$

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

This conjecture gives a quantitative strengthening of the uniform boundedness of the two diagonal-edge invariants over positive convex structures with bounded canonical area. The supplied text does not indicate whether the conjecture has been resolved.