Core-structure conjecture for conditionally non-redundant hypergraphs

Let $H=(V,E)$ be a hypergraph of arity $5$ that is conditionally non-redundant for $P_{181}\mid Q_{181}$. For any vertex $v\in V$, let $H_v=(V,E_v)$ be the hypergraph whose hyperedges contain $v$. If $|E_v|=\omega(n)$, then there exists another vertex $u\in V$ such that

$$(u,v)\subseteq e$$

for every $e\in E_v$.

Core-structure conjecture. Under these hypotheses, every vertex incident with more than linearly many hyperedges shares a second common vertex with all those hyperedges.

This is proposed as an analog of a structural result of Deza, Erdős, and Frankl for hypergraphs with restricted intersection sizes. The source presents the analog as false, so the claim is refuted in the paper's discussion.