  Athanasiadis conjectured that, for every positive integer $r$, the local
$h$-polynomial of the $r$th edgewise subdivision of any simplex has only real
zeros. In this paper, based on the theory of interlacing polynomials, we prove
that a family of polynomials related to the desired local $h$-polynomial is
interlacing and hence confirm Athanasiadis' conjecture.
