Tightness of the random-order selectability ratio for k-matchoids

Determine whether the random-order competitive ratio \((1-e^{-k})/k\) is tight for \(k\)-matchoids or, more specifically, for intersections of \(k\) matroids.

Background

The paper proves a $(1-e^{-k})/k$-ex-ante prophet inequality and an associated random-order contention resolution scheme for $k$-matchoids. The authors note that a strictly better ratio is known for $k$-bounded hypergraph matching, which is a special case of the broader $k$-matchoid framework. Consequently, it remains unresolved whether the ratio established in the paper is optimal for all $k$-matchoids, or even for the narrower class of intersections of $k$ matroids.