The weight-six generation conjecture for integral modular-form rings

Let $N\geq 1$. Write $M(\Gamma_0(N),\mathbb{Z}[\frac{1}{6N}])$ for the graded algebra of modular forms for $\Gamma_0(N)$ with coefficients in $\mathbb{Z}[\frac{1}{6N}]$. Weight-six generation conjecture. For any $N\geq 1$, the $\mathbb{Z}[\frac{1}{6N}]$-algebra $M(\Gamma_0(N),\mathbb{Z}[\frac{1}{6N}])$ is generated in weight at most $6$. Numerical evidence appears to support this conjecture, which predicts a uniform bound on the weights needed to generate the rings of modular forms for all congruence subgroups $\Gamma_0(N)$.