Simon's conjecture on shelling completable complexes

Let $\Delta$ be a pure simplicial complex. It is shellable if its facets admit an ordering satisfying the condition that, at each step, the intersection of the complex formed by the preceding facets with the new facet is pure of codimension one. It is shelling completable if some shelling of $\Delta$ is an initial sequence of a shelling of the full simplex skeleton of the same dimension on the same vertex set.

Simon's conjecture. Every shellable complex is shelling completable.

This conjecture concerns whether every shelling can be extended, in the sense that some shelling of a shellable complex occurs as the beginning of a shelling of the corresponding skeleton of a simplex. The source provides no resolution status.