Full-rank conjecture for descendent matroids

For each weight $k$, let $\mathcal{M}_k$ be the descendent matroid whose ground set consists of stationary descendents of weight $k$, and let $\operatorname{QM}_k$ be the space of quasimodular forms of weight $k$. The rank of a matroid is the cardinality of any basis, and $\operatorname{dim}(\operatorname{QM}_k)$ denotes the dimension of this quasimodular-form space. The conjecture also uses the stationary descendents $\langle\tau_{k_1}\ldots\tau_{k_n}\rangle$ with $\sum_i(k_i+2)=k$.

Full-rank conjecture. The rank of $\mathcal{M}_k$ is always equal to $\operatorname{dim}(\operatorname{QM}_k)$. Equivalently,

$$\mathbb{Q}\left[\left\{\left\langle\tau_{k_1}\ldots\tau_{k_n}\right\rangle\right\}_{\sum(k_i+2)=k}\right]=\operatorname{QM}_k.$$

This conjecture would ensure that stationary descendents span the full space of quasimodular forms of each weight and, in particular, that the relevant bases exist. The paper reports computations supporting its plausibility but does not establish it in general.