This question is a follow-up of question A series that is rational? . Let $k=\mathbb F_q(T)$. Can one prove (or disprove) that the series $\sum_{n\ge0}(1-TX^{q^n})Y^{q^n}\in k[[X,Y]]$ is algebraic over $k(X,Y)$? In the mentioned link it is proved that it does not belong to $k(X,Y)$.

If the series would be $\sum_{n\ge0}(1-X^{q^n})Y^{q^n}$, it is easy to prove...