Strong vertex decomposability of infinite-type spherical subword complexes

Let $\mathcal{S}(Q,\pi)$ be a spherical subword complex of infinite type. Strong vertex-decomposability conjecture. The complex $\mathcal{S}(Q,\pi)$ is strongly vertex decomposable. This is equivalent in the paper to the conjecture that deleting any vertex from an infinite-type spherical subword complex produces a vertex-decomposable complex, and would extend the finite-type theorem to all types.