Integral generators conjecture for square-free moonshine groups

Let $N$ be square-free, let $g>0$ be the genus of the moonshine modular curve associated to the group $\Gamma_0(N)^+$, and consider its function field. Integral generators conjecture. This function field admits two generators whose $q$-expansions have integer coefficients after the lead coefficient has been normalized to equal one, and the orders of their poles at $i\infty$ are at most $g+2$. This conjecture extends the authors' results on integral $q$-expansions and small pole orders to all square-free levels, independently of the gap sequence at $i\infty$; the genus-one case follows from known results, while the genus-two and genus-three cases were analyzed separately in the paper.