Find a counterexample to the universal \(n+m\) minimal-width formula

Construct a pair of integers \(n,m\) with \(1\leq m\leq n\) for which the minimal width of ReLU networks is strictly less than \(n+m\), or otherwise determine whether the equality \(w_{\mathrm{min}(n,m)}=n+m\) holds throughout this range.

Background

For ReLU networks, the paper proves that the minimal width equals $n+m$ when $n\in\{2,3,4,7,8\}$ and $1\leq m\leq n$, using smooth immersions of real projective spaces. The authors state that they do not expect this formula to hold for every pair in the broader range $1\leq m\leq n$, but they do not provide a counterexample. They note that the equality is already known to fail when $m>2n\geq4$, leaving the stated range as an unresolved question.