The Lipschitz-surface characterization conjecture for $d$-controlling sequences

Let $m$ and $d$ be positive integers, and let $(x_i)_{i\in I}$ be a sequence of points in $\mathbb{R}^m$. A sequence in $\mathbb{R}^m$ is $d$-controlling if there are points $(y_i)_{i\in I}\subset\mathbb{R}^d$ such that every Lipschitz function $f:\mathbb{R}^m\to\mathbb{R}^d$ satisfies $|f(x_i)-y_i|<1$ for some $i\in I$.

The Lipschitz-surface characterization conjecture. The sequence $(x_i)_{i\in I}$ is $d$-controlling if and only if there exist a Lipschitz map $g:\mathbb{R}^d\to\mathbb{R}^m$ and a $d$-controlling sequence $(x'_i)_{i\in I'}$ in $\mathbb{R}^d$, with $I'\subseteq I$, such that

$$g(x'_i)=x_i\qquad(i\in I').$$

The claim is intended to characterize controlling sequences by a $d$-dimensional Lipschitz surface through a controlling subset. It is stated as obviously true when $m\le d$; the case $m>d$ is the substantive open case.