Circuit decomposition conjecture for Eulerian binary matroids

Let $M$ be an Eulerian binary matroid, meaning a subset of $\mathbb F_2^n\setminus\{0\}$ whose elements can be decomposed into circuits. Write $c(M)$ for the minimum number of circuits in a circuit decomposition of $M$, and let $\operatorname{rank}(M)$ denote its rank. Circuit decomposition conjecture. For every Eulerian binary matroid $M\subseteq\mathbb F_2^n\setminus\{0\}$,

$$c(M)\leq \left\lceil\frac{2^{\operatorname{rank}(M)}-1}{\operatorname{rank}(M)+1}\right\rceil.$$

This would extend, up to rounding, the established bound for complete binary matroids to arbitrary Eulerian binary matroids; the conjecture remains open.