Height bound conjecture for modular polynomials with level structures

Let $N$ be the level parameter, let $\alpha$ be the invariant defining the modular polynomials with level structures, and let $J_1$ be the corresponding polynomial. For a polynomial $F(X,Y)=\sum_{i,j}c_{i,j}X^iY^j\in\mathbb{Z}[X,Y]$, define its logarithmic height by

$$h(F)=\max_{i,j}\log|c_{i,j}|,$$

with $\log 0=0$. Let $B_\ell=6\ell\log\ell+16\ell+\min\{2\ell,14\sqrt{\ell}\log\ell\}$ be the stated upper bound for the height of the classical modular polynomial $\Phi_\ell$, and let $\Phi_\ell^{\alpha}$ denote the corresponding modular polynomial with level structure. Height bound conjecture. There exists a bound $L$ depending on $\alpha$ such that, for any prime number $\ell>L$ not dividing $N$,

$$h(\Phi_\ell^{\alpha})\leq B_\ell/\deg(J_1).$$

This conjecture naturally generalizes a heuristic for classical modular polynomials and is motivated by comparing the logarithmic heights in a resultant identity relating $\Phi_\ell^{\alpha}$ to $\Phi_\ell$.