Hypergraph reconstruction conjecture from hyperedge-deleted subhypergraphs

Let $\mathcal{H}_{\operatorname{yper}}=(\Lambda,\mathcal{E})$ and $\mathcal{H}^*_{\operatorname{yper}}=(\Lambda^*,\mathcal{E}^*)$ be hypergraphs based on 3I-hyperedge sets $\mathcal{E}\in\mathcal{E}(\Lambda^2)$ and $\mathcal{E}^*\in\mathcal{E}((\Lambda^*)^2)$, with $|\mathcal{E}|=|\mathcal{E}^*|$. Hypergraph isomorphism conjecture. If there is a bijection $\theta:\mathcal{E}\rightarrow\mathcal{E}^*$ such that, for each $e\in\mathcal{E}$, the hypergraphs obtained by deleting $e_{\cap}$ and $e$ are isomorphic,

$$\bigl(\Lambda\setminus\{e_{\cap}\},\mathcal{E}\setminus e\bigr)\cong\bigl(\Lambda^*\setminus\{\theta(e_{\cap})\},\mathcal{E}^*\setminus\theta(e)\bigr),$$

where $e_{\cap}\subset e\in\mathcal{E}$ and $e_{\cap}\cap e'=\emptyset$ for every $e'\in\mathcal{E}\setminus e$, then $\mathcal{H}_{\operatorname{yper}}\cong\mathcal{H}^*_{\operatorname{yper}}$. This is a hypergraph analogue of reconstruction from deleted substructures; the source gives no resolution status.