The 11.8 lower bound for the asymptotic height of classical modular polynomials

For each prime $l$, let $\Phi_l$ be the classical modular polynomial and let $h(\Phi_l)$ denote its height. Consider the normalized excess of its height over the leading term $6l\log l$.

Asymptotic height lower-bound conjecture. For prime $l$,

$$\liminf_{l\to\infty}\frac{h(\Phi_l)-6l\log l}{l}>11.8.$$

The conjecture asserts that the linear correction term in the height remains asymptotically bounded below by a constant strictly greater than $11.8$. The supplied text gives numerical motivation from tabulated values but no resolution.