Vector-valued Sprecher representation conjecture

Let $n,m\in\mathbb{N}$ with $m>1$, and let $f:[0,1]^n\to\mathbb{R}^m$ be continuous. A Sprecher Network has architecture $n\to[d_1]\to m$ with one hidden block of width $d_1\geq 2n+1$ and an output block, using continuous splines $\phi^{(1)},\Phi^{(1)},\phi^{(2)},\Phi^{(2)}$, monotonic $\phi^{(1)}$ and $\phi^{(2)}$, suitable parameters $\lambda^{(1)},\eta^{(1)},\lambda^{(2)},\eta^{(2)}$, and optionally lateral mixing parameters. Vector-valued Sprecher representation conjecture. For every $\epsilon>0$, there exists such a network whose output $\hat f(\mathbf{x})$ satisfies

$$\sup_{\mathbf{x}\in[0,1]^n}\|f(\mathbf{x})-\hat f(\mathbf{x})\|_{\mathbb{R}^m}<\epsilon.$$

This extends the scalar Sprecher representation to vector-valued functions, but the supplied context states that universality of this specific construction is not directly covered by Sprecher's theorem. The status of the vector-valued extension therefore remains open.