Lovász–Mason flat-condition conjecture for matroid symmetric powers

Let $M$ be a matroid with ground set $E$, let $(M_1,\dots,M_d)$ be a symmetric quasi power of $M$, and let $\sF(M)$ denote the set of flats of $M$. For each $F\in\sF(M)$, write $F\cdot\operatorname{Sym}_{d-1}(E)$ for the corresponding subset of the ground set of $M_d$. Lovász–Mason flat-condition conjecture. If

$$F\cdot\operatorname{Sym}_{d-1}(E)\in\sF(M_d)$$

for every $F\in\sF(M)$, then $M_d$ has rank

$$\binom{\operatorname{rk}(M)+d-1}{d}.$$

This is the reverse direction of the implication from the symmetric-power axioms to the flat condition proposed by Mason. The implication from the stronger symmetric-power axiomatization to the flat condition is proved in the paper, but the converse remains open despite substantial effort; the problem was left open by Lovász and Mason.