Necessary-and-sufficient closedness conjecture for shallow sparse ReLU networks

Let $\Omega=[-B,B]^{N_0}$ with $B>0$, and let $\boldsymbol{I}=(I_2,I_1)$ be a shallow-network architecture. Let $\mathcal{F}_{\boldsymbol{I}}(\Omega)$ be its function space. The function space is considered with the uniform norm in

$$(C^0(\Omega),\|\cdot\|_\infty).$$

Shallow-network closedness conjecture. The set $\mathcal{F}_{\boldsymbol{I}}(\Omega)$ is closed in $(C^0(\Omega),\|\cdot\|_\infty)$ if and only if $\boldsymbol{I}$ satisfies the second condition of the stated generalized closedness theorem.

This conjecture seeks a necessary and sufficient condition for closedness in the shallow case $L=2$, where the paper has separately obtained necessary and sufficient conditions. The source does not state a resolution.