Non-archimedean Broyden convergence conjecture over formal power series

Let the hypotheses and notation be those of Theorem, and work over the formal power-series field $\mathbb{Q}\llbracket T \rrbracket$. For Broyden's method, set

$$u_n=\frac{s_n}{s_n^t\cdot s_n}.$$

Formal-power-series Broyden convergence conjecture. Broyden's method with $\nu_n=\frac{s_n}{s_n^t\cdot s_n}$ has locally Q-superlinear convergence. This is proposed as an adaptation of the preceding non-archimedean convergence conjecture to $\mathbb{Q}\llbracket T \rrbracket$, where the relevant norm identities still do not yield the real proof.