Semi-specializability of Chebyshev polynomial products for multiples of four

Let $T_m(x)$ denote the Chebyshev polynomial of index $m$, and let $\ell$ be a non-zero multiple of $4$. Consider

$$\prod_{n=0}^{\infty}\left(1+\frac{1}{T_{\ell^n}(x)}\right).$$

A continued fraction is semi-specializable if it has the partial specialization property defined in the paper, without necessarily being fully specializable. Chebyshev product conjecture. The product has a semi-specializable continued fraction expansion. The preceding discussion observes semi-specializability for $\ell\equiv0\pmod 4$ but does not establish it; the claim concerns the anticipated general pattern for these Chebyshev products.