Why does this paper claim to have found a minimal width of $d_{in}+1$?
From the abstract of the paper:

Specifically, we answer the following question: for a fixed $d_{in} \ge 1$,
what is the minimal width $w$ so that neural nets with ReLU
activations, input dimension $d_{in}$, hidden layer widths at most $w$,
and arbitrary depth can approximate any continuous, real-valued
function of $d_{in}$ in variables arbitrarily well?

&quot;Real-valued function&quot; means that the output dimension $d_{out}=1$. Their Theorem 1 is more general and says that for any $d_{in}, d_{out} \ge 1$ you will need $d_{in}+1 \le w \le d_{in}+d_{out}$.