The level-three and level-four binary-index conjecture for MPVs

Let $N$ be a level, let $L(\{1\}^w\mid t_1,\dots,t_w)$ denote an MPV of weight $w$ with all indices equal to $1$, and let $d(w,N)$ be the $\Q$-dimension of the weight-$w$ MPVs of level $N$.

Binary-index conjecture. If $N=3$ or $N=4$, every MPV of level $N$ is a linear combination of MPVs

$$L(\{1\}^w\mid t_1,\dots,t_w),\qquad t_j\in\{1,2\},$$

and consequently

$$d(w,N)=2^w\qquad\text{for all }w\geq 1.$$

This gives the expected dimension and a binary-index spanning family at levels three and four; the source says that computations in small weights support it.