Simon's conjecture on extendable shellability of uniform matroid independence complexes

For integers $1<k\le n$, let $U_{n,k}$ be the uniform matroid and let its independence complex be the simplicial complex whose facets are the bases of $U_{n,k}$. A partial shelling is an ordering $B_1<\dots<B_s$ of bases that shells the subcomplex generated by those bases. Simon's conjecture. For every $1<k\le n$, every such partial shelling can be extended: if $B_1<\dots<B_s$ shells $\langle B_1,\dots,B_s\rangle$, then the remaining bases can be ordered as $B'_1<\dots<B'_t$ so that

$$B_1<\dots<B_s<B'_1<\dots<B'_t$$

is a shelling order of the independence complex of $U_{n,k}$. The source explains that this conjecture is widely believed to be false, notes counterexamples to extendable shellability for other matroid independence complexes, and says that proving or disproving it appears to require a genuinely novel idea.