Merino's Lissajous decomposition conjecture for Chebyshev 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. The associated real plane curve is defined by

$$T_n^2(x)-2T_n(x)T_m(y)\cos\delta+T_m^2(y)-\sin^2\delta=0.$$

Merino's conjecture. This curve is the union of a finite number of Lissajous curves. The conjecture concerns the decomposition of the Chebyshev-defined curve into parametrized Lissajous components; its status is not established by the supplied text.