Regular closedness off the real axis for root closures of bounded-height integer polynomials

For each integer $h\ge1$, let $Z_h$ be the closure in the complex plane of the roots of nonzero integer polynomials whose coefficients have absolute value at most $h$, with no degree restriction. We prove that $Z_h$ is regular closed off the real axis: every non-real point of $Z_h$ is a limit of int