Congruence conjecture for the normalized generating function of vacuum modules

Let $L(k\Lambda_0)$ denote the level-$k$ vacuum module, let $\mathcal{F}_{L(k\Lambda_0)}(q)$ be its normalized generating function, and let $p=2k+3$ be a prime with $p\geq 5$. Congruence conjecture. For every $k\in\mathbb{N}$,

$$\mathcal{F}_{L(k\Lambda_0)}(q)\equiv 1\pmod p.$$

This congruence is proposed on the basis of numerical evidence and concerns the $p$-integrality properties of the generating function.