Kajitani–Ueno–Miyano's Hamiltonian matroid conjecture

Let $M=(E,\mathcal{I})$ be a matroid with rank function $r$. It is Hamiltonian if some cyclic ordering of $E$ has the property that every segment of $r(M)$ consecutive elements is a basis of $M$. It is uniformly dense if, for every nonempty subset $X\subseteq E$,

$$\frac{|X|}{r(X)}\leqslant\frac{|E|}{r(E)}.$$

Kajitani–Ueno–Miyano's conjecture. A matroid $M$ is Hamiltonian if and only if it is uniformly dense. Hamiltonian matroids are known to be uniformly dense, while the converse is stated as a conjecture and is not resolved in the supplied text.