Conjecture relating function-space and finite-set closedness for sparse ReLU networks

Let $B>0$, let $\boldsymbol{I}$ be an architecture, and let $\mathcal{F}_{\boldsymbol{I}}(\Omega)$ denote its function space on a domain $\Omega$. The function space of $\boldsymbol{I}$ is closed in

$$(C^0([-B,B]^d),\|\cdot\|_\infty),$$

if and only if $\mathcal{F}_{\boldsymbol{I}}(\Omega)$ is closed for every finite set $\Omega$.

Closedness-equivalence conjecture. An architecture $\boldsymbol{I}$ has a closed function space on the cube if and only if its realization set is closed on arbitrary finite domains.

If true, results for the cube domain and finite-set domains would imply one another. The cube case permits tools such as Jacobian matrices and continuity, while finite-set closedness is more directly connected to practical neural-network training problems. The source does not state a resolution.