Lovász–Mason contraction reduction for matroid symmetric powers

Let $M$ be a matroid on the ground set $E$, let $(M_1,\dots,M_d)$ be a symmetric quasi power of $M$, and suppose that $F\cdot\operatorname{Sym}_{d-1}(E)$ is a flat of $M_d$ for every flat $F$ of $M$. Lovász–Mason contraction reduction. There is an element $e\in E$ such that the contraction

$$M_d\mathbin{\contr}\bigl(e\cdot\operatorname{Sym}_{d-1}(E)\bigr)$$

is a symmetric quasi power of $M\mathbin{\contr}e$. This is presented as an auxiliary result sufficient to prove the Lovász–Mason flat-condition conjecture. Its resolution is not supplied in the given text, so it remains open here.