The bounded-weight Schur MacMahon spanning conjecture

Let $F_{\leq k}\mathcal{M}_{\mathbb{Z}}:=F_{\leq k}\mathcal{M}\cap\mathbb{Z}[[q]]$ be the integral quasimodular forms of filtered weight at most $k$, and let $g(\lambda)$ be the Schur MacMahon series indexed by partitions. Bounded-weight spanning conjecture. There is an explicit function $B(k)$ such that for every $k\geq0$,

$$F_{\leq k}\mathcal{M}_{\mathbb{Z}}\subseteq\sum_{|\lambda|\leq B(k)}\mathbb{Z}\,g(\lambda).$$

This is a finite, weight-bounded refinement of the integral-lattice conjecture. The depth-one sector is established, but the stated bounded-weight inclusion is presented as a conjectural refinement.