The integral-lattice conjecture for Schur MacMahon series

Let $\mathcal{M}_{\mathbb{Z}}:=\mathcal{M}\cap\mathbb{Z}[[q]]$ be the integral lattice of quasimodular forms, and let $\mathcal{G}_{\mathbb{Z}}:=\sum_\lambda\mathbb{Z}\,g(\lambda)$ be the subring generated by the Schur MacMahon series. Integral-lattice conjecture. The Schur MacMahon series span the integral lattice over $\mathbb{Z}$, i.e.

$$\mathcal{G}_{\mathbb{Z}}=\mathcal{M}_{\mathbb{Z}}.$$

This refines the rational spanning result by asking for an integral basis-generating statement; the inclusion in the displayed equality is known, while the reverse inclusion remains open.