Tchebyshev transform conjecture for spherical shellable posets

Let $P$ be a poset that is spherical and shellable, and let $T(P)$ denote its Tchebyshev transform. Spherical shellability conjecture. If $P$ is spherical and shellable, then $T(P)$ is also spherical and shellable. This would extend Hetyei's theorem from face posets of spherical $CW$-complexes to all spherical shellable posets; the source gives no resolution of the conjecture.