Divisibility conjecture for exponents of modular-unit basis elements

Let $N$ be a positive integer, let $L$ be the level-dependent integer used for the Galois action, and let $\mathcal{D}_N$ and $F_{m,h}$ be the indexing set and modular-unit basis introduced in the paper. Let $\ell(m)$ be the associated integer and let $\varphi$ denote Euler's totient function. Suppose $n\geq 2$ and

$$F=\prod_{m\in\mathcal{D}_N}\prod_{h=0}^{\varphi(\ell(m))-1}F_{m,h}^{e(m,h)},\qquad e(m,h)\in\mathbf{Z},$$

is a modular unit on $X_0(N)$. Assume that the order of $F$ at every cusp is divisible by $n$, and that $(\sigma_s(F)/F)^{1/n}$ is a modular unit on $X_0(N)$ for every integer $s$ coprime to $L$. The divisibility conjecture. Then $e(m,h)$ is divisible by $n$ for every $m\in\mathcal{D}_N^*$ and every $h\ne0$. This is proposed as the key divisibility statement underlying the strategy for Yoo's conjecture; the paper does not provide a general proof.