Zero-location conjecture for the normalized generating function of vacuum modules

Let $L(k\Lambda_0)$ be the level-$k$ vacuum module, let $\mathcal{F}_{L(k\Lambda_0)}$ be its normalized generating function, and let $G(\mathcal{F}_{L(k\Lambda_0)},j)$ be the polynomial obtained from the modular-form decomposition

$$f(\tau)=\Delta^t(\tau)E_4^\delta(\tau)E_6^\epsilon(\tau)G(f,j(\tau)),$$

where

$$j(\tau)=\frac{1728E_4^3(\tau)}{E_4^3(\tau)-E_6^2(\tau)}.$$

Zero-location conjecture. For every $k\neq 2i^2-2$, $i\geq 2$, the zeros of $G(\mathcal{F}_{L(k\Lambda_0)},j)$ are simple and lie inside the interval $[0,1728]$. This is a numerical conjecture about the zeros of the polynomial associated with the modular form; via the stated parametrization of the fundamental-domain arc by the $j$-function, it describes the location of the corresponding zeros of the generating function.