Kajitani–Miyano–Ueno conjecture on cyclically orderable matroids

Let $M$ be a matroid with ground set $E(M)$ and rank function $r$. A matroid is cyclically orderable if it has a cyclic permutation of its elements such that any $r(M)$ consecutive elements form a basis. For every nonempty subset $X\subseteq E(M)$, define

$$\beta(X)=\frac{|X|}{r(X)}$$

when $r(X)\ne 0$, and define $\beta(X)=\infty$ otherwise. Let

$$\gamma(M)=\max_{\mathchar"001F\ne X\subseteq E(M)}\beta(X).$$

Kajitani–Miyano–Ueno conjecture. The matroid $M$ is cyclically orderable if and only if

$$\gamma(M)=\beta(E(M)).$$

Equivalently, this asserts that cyclic orderability is characterized by $|X|/r(X)\le |E(M)|/r(M)$ for every nonempty $X\subseteq E(M)$, with the stated convention when $r(X)=0$. The paper verifies the conjecture for all paving matroids, but the general case is not resolved here.