Jointly scattered derivative conjecture for analytic compositions

Let $I\subset\mathbb{R}$ be a compact interval and let $k\in\mathbb{N}$. For each $i=1,\dots,k$, let $(u_{i,n})_{n\in\mathbb{N}}$ be a sequence of functions in $C^1(I,\mathbb{R})$ whose derivatives $u_{i,n}'$ are monotone for every $i$ and $n$. Let $f_1,\dots,f_k:\mathbb{R}\to\mathbb{R}$ be analytic functions, and assume that for every $a\in\mathbb{R}$ the sequences $u_{1,n}'(a),\dots,u_{k,n}'(a)$ are jointly scattered. Jointly scattered derivative conjecture. For almost every $x\in\mathbb{R}$, the sequence

$$\big(u_{1,n}(f_1(x)),\dots,u_{k,n}(f_k(x))\big)_{n\in\mathbb{N}}$$

is uniformly distributed in $\mathbb{T}^k$. This is the second proposed way to avoid the counterexample in the preceding discussion; the paper gives no resolution.