On t-adic Littlewood conjecture for generalised Thue-Morse functions

We consider a Laurent series defined by infinite products $g_u(t) = \prod_{n=0}^\infty (1 + ut^{-2^n})$, where $u\in \mathbb{F}$ is a parameter and $\mathbb{F}$ is a field. We show that for all $u\in\mathbb{Q}\setminus\{-1,0,1\}$ the series $g_u(t)$ does not satisfy the $t$-adic Littlewood conjecture. On the other hand, if $\mathbb{F}$ is finite then $g_u(t)\in \mathbb{F}((t^{-1}))$ is either a rational function or it satisfies the $t$-adic Littlewood conjecture.