Odd-cover conjecture for Eulerian binary matroids

Let $M$ be an Eulerian binary matroid, viewed as a subset of $\mathbb F_2^n\setminus\{0\}$. Let $c_2(M)$ be the minimum size of an odd-cover of $M$, and let $a(M)$ be the minimum number of linearly independent sets whose union is $M$. Odd-cover conjecture. For every Eulerian binary matroid $M\subseteq\mathbb F_2^n\setminus\{0\}$,

$$c_2(M)\leq a(M).$$

The paper proves the asymptotic relation $c_2(M)=(1+o(1))a(M)$ when $a(M)\to\infty$ and gives examples attaining the lower bound $a(M)$; the claimed exact upper bound remains open.