Linusson's bunkbed conjecture for hypergraph percolation

Let $\mathcal{H}$ be a hypergraph and let $T\subseteq V(\mathcal{H})$ be a set of posts. In the conditioned model $E_4^T(\widetilde{\mathcal{H}})$, precisely one of the two copies of each hyperedge is retained, and precisely the vertical edges at vertices in $T$ are retained. For vertices $u,v\in V(\mathcal{H})$, write $\operatorname{\mathbb{P}}(u^{(0)}\xleftrightarrow{E_4^T(\mathcal{H})}v^{(i)})$ for bidirectional connectivity. Linusson's hypergraph bunkbed conjecture. For every hypergraph $\mathcal{H}$ and every $T\subseteq V(\mathcal{H})$,

$$\operatorname{\mathbb{P}}(u^{(0)}\xleftrightarrow{E_4^T(\mathcal{H})}v^{(0)})\geq \operatorname{\mathbb{P}}(u^{(0)}\xleftrightarrow{E_4^T(\mathcal{H})}v^{(1)}).$$

This conjecture is disproved in the paper by an explicit hypergraph example.