$σ$-hereditarily closure-preserving $\mathcal I$-$sn$-networks, $\mathcal I$-$sn$-metrizability, and local HCP-finiteness

Let $\mathcal I$ be an admissible ideal on $\mathbb N$. Zhou, Liu, Liu, and Lin asked whether every regular space with a $σ$-hereditarily closure-preserving $\mathcal I$-$sn$-network is $\mathcal I$-$sn$-metrizable. The answer is affirmative for every admissible ideal. The proof rests on a local cou