  Athanasiadis raised the question whether the local $h$-polynomials of type
$A$ cluster subdivisions have only real zeros. In this paper, we confirm this
conjecture and prove the real-rootedness of local $h$-polynomials for all the
other Cartan--Killing types. Our proofs mainly involve multiplier sequences and
Chebyshev polynomials of the second kind.

Combinatorics