Recursive formula conjecture for level-2 generalized $(q,t)$-Catalan numbers

Let $C_n^{(2)}(q,t)$ denote the generalized $(q,t)$-Catalan number of level $2$. Level-2 recursion conjecture. For $n>1$,

$$\begin{cases} C^{(2)}_{n+1}(q,t)=t^n C^{(2)}_n(q/t,t) & \text{if $n$ is even},\\ C^{(2)}_{n+1}(q,t)=t^n C^{(2)}_n(q,t)+qt^{n-1}C^{(2)}_{n-1}(q,t) & \text{if $n$ is odd}. \end{cases}$$

The formula is presented as a conjectural special-case description of the level-2 generalized Catalan numbers, and the source provides no proof or resolution.