Uniform-minor conjecture for positive descendent matroids

For each weight $k$, let $S^k$ be the set of positive stationary descendents of weight $k$, and let $\mathcal{M}_k\mid S^k$ denote the restriction of the descendent matroid $\mathcal{M}_k$ to $S^k$. Write $\operatorname{QM}_k$ for the space of quasimodular forms of weight $k$, and let $U_{r,n}$ denote the uniform matroid of rank $r$ on $n$ elements. A matroid has $U_{r,n}$ as a minor if it can be obtained from it by deletions and contractions.

Uniform-minor conjecture. For every $k\geq 14$, there exists an integer $n>\operatorname{dim}(\operatorname{QM}_k)$ such that $\mathcal{M}_k\mid S^k$ has $U_{\operatorname{dim}(\operatorname{QM}_k),n}$ as a minor.

The conjecture predicts a systematic uniform-matroid structure after restricting to positive stationary descendents. The paper gives examples in weights $14$, $16$, and $18$, while the general assertion remains open.