Merino's irreducibility conjecture for Chebyshev-Lissajous polynomials

Let $m$ and $n$ be positive integers, let $\delta$ be a real number with $\sin\delta\neq 0$, and let $T_k$ denote the $k$th Chebyshev polynomial. Consider the polynomial

$$T_n^2(x)-2T_n(x)T_m(y)\cos\delta+T_m^2(y)-\sin^2\delta.$$

Merino's irreducibility conjecture. This polynomial is irreducible over $\mathbb{R}$ if and only if $m$ and $n$ are coprime. The claim gives a precise arithmetic criterion for irreducibility of these Chebyshev-Lissajous polynomials; the supplied text does not establish its resolution.