Koksma's analytic composition conjecture

Let $(u_n)_{n\in\mathbb{N}}$ satisfy the conditions of the cited Koksma theorem, and let $f_1,\dots,f_k:\mathbb{R}\to\mathbb{R}$ be analytic functions such that $1,f_1,\dots,f_k$ are linearly independent. Koksma's analytic composition conjecture. For almost every $x\in\mathbb{R}$, the sequence

$$\big(u_n(f_1(x)),\dots,u_n(f_k(x))\big)_{n\in\mathbb{N}}$$

is uniformly distributed in $\mathbb{T}^k$. This is proposed as a curve version of Koksma's theorem; the preceding example shows why linear independence alone requires care, and the claim remains open.