A 12l upper bound for the height of the classical modular polynomial

Let $l$ be a prime, and let $\Phi_l$ denote the classical modular polynomial of prime level $l$. Its height $h(\Phi_l)$ is the logarithm of the maximum absolute value of its coefficients.

Height upper-bound conjecture. For every prime $l>30$,

$$h(\Phi_l)<6l\log l+12l.$$

This would improve the proven bound $h(\Phi_l)\leq 6l\log l+18l$ and is motivated by the numerical data in the paper; its status is not resolved in the supplied text.