Suppose I have a set of hyperplanes $\{H_{i}\}$ of a (finite) matroid $\mathcal{M}$ the smallest modular cut over which is non-proper. It doesn’t seem to be true in general that there exists an ordering of a subset of length $r$ of these hyperplanes $H_{\sigma(i)}$ such that for all $k\le r$, $\cap_{i<k}H_{\sigma(i)}$ and $H_{\sigma(k)}$ are modular, but is there a slightly more general nice condition in this vein which is equivalent to the $H_{i}$ generating the non proper modular cut of $\mathcal{M}$, in that it looks at modular intersections of hyperplanes?
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