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

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

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

Continued-fraction conjecture. The positive powers of $q$ follow the sequence $0,3,10,21,\dots$, given by $(k-1)(2k-1)$, while the negative powers follow $0,5,14,27,\dots$, given by $(k-1)(2k+1)$.

This conjecture predicts a further continued-fraction pattern for the q-tangent function. The supplied text gives no proof or resolution, so the conjecture remains open.