Existence of a self-reciprocal polynomial with certain root properties

Question

Does there exist a monic self-reciprocal integer polynomial $p\in\mathbb{Z}[x]$ of degree 10 with roots $r\textrm{e}^{\pm\textrm{i}\theta_1}, r^{-1}\textrm{e}^{\pm\textrm{i}\theta_1}, r\textrm{e}^{\pm\textrm{i}\theta_2}, r^{-1}\textrm{e}^{\pm\textrm{i}\theta_2}, r,$ and $r^{-1}$ with the following properties:

1. modulus $r>1$, and
2. no quotient of two distinct roots of $p$ is a root of unity