Generator-weight conjecture for integral modular forms on $\Gamma_0(p)$

Let $M(\Gamma_0(p),\mathbb{Z})$ be the graded algebra of integral modular forms on $\Gamma_0(p)$, and let $T\in M_{p-1}(\Gamma_0(p),\mathbb{Z})$ be the $T$-form. Generator-weight conjecture. The weights of the modular forms in a minimal generating set for $M(\Gamma_0(p),\mathbb{Z})$ belong to

$$\{2,4,6,p-1\},$$

and there is exactly one generator of weight $p-1$, which may be chosen to be $T$. This is presented as a consequence of the weight-six generation conjecture together with the proved generation theorem; the supplied text gives numerical evidence for the underlying conjecture but does not state a resolution.