The q-tangent continued-fraction conjecture for parameter tuple $(1,0,1,0)$

Let $q$ be a parameter and let $\tan_q(z)$ denote the q-tangent function. For the parameter tuple $(A,B,C,D)=(1,0,1,0)$, consider the continued fraction

$$\tan_q(z)=\cfrac{z}{[1]_q q^{0}-\cfrac{z^2}{[3]_q q^{-2}-\cfrac{z^2}{[5]_q q^{1}-\cfrac{z^2}{[7]_q q^{-9}-\cfrac{z^2}{\ddots}}}}}.$$

Continued-fraction conjecture. The positive powers of $q$ in this continued fraction follow the sequence $0,1,6,15,\dots$, given by $k(2k-1)$, while the negative powers follow $2,9,20,35,\dots$, given by $(k+1)(2k-1)$.

This conjecture proposes an explicit continued-fraction pattern for the q-tangent function in the indicated parameter case. The associated continuant polynomials are not known: the source states that the coefficients cannot presently be guessed and leaves their expansion as an open problem.