Polynomial comparability conjecture for exhaustion subsets of convex structures

Let $\mathbf{L}\in\mathbb{R}_{>0}^{2m}$, and let $A^t(S_{g,m})(\mathbf{L})$, $B^t(S_{g,m})(\mathbf{L})$, $C^t(S_{g,m})(\mathbf{L})$, $D^t(S_{g,m})(\mathbf{L})$, $E^t(S_{g,m})(\mathbf{L})$, $F^t(S_{g,m})(\mathbf{L})$, $G^t(S_{g,m})(\mathbf{L})$, $\mathcal{H}^t(S_{g,m})(\mathbf{L})$, and $\mathcal{AH}^t(S_{g,m})(\mathbf{L})$ be the subsets of $\mathcal{H}(S_{g,m})(\mathbf{L})$ appearing in the source's exhaustion construction. Here, subsets are polynomially comparable when the comparison parameters are related by polynomial functions of $t$.

Polynomial comparability conjecture. The following subsets of $\mathcal{H}(S_{g,m})(\mathbf{L})$ are polynomially comparable to each other:

$$A^t(S_{g,m})(\mathbf{L}),\quad B^t(S_{g,m})(\mathbf{L}),\quad C^{\log t}(S_{g,m})(\mathbf{L}),\quad D^{\frac{t}{t-1}}(S_{g,m})(\mathbf{L}),\quad E^t(S_{g,m})(\mathbf{L}),$$ $$F^t(S_{g,m})(\mathbf{L}),\quad G^t(S_{g,m})(\mathbf{L}),\quad \mathcal{H}^t(S_{g,m})(\mathbf{L}),\quad \mathcal{AH}^t(S_{g,m})(\mathbf{L}).$$

The preceding proposition establishes ordinary comparability among several of these subsets, while the conjecture asks for the sharper polynomial dependence, including the explicitly displayed exponents. The supplied text does not indicate whether this quantitative conjecture has been resolved.