Antichain k-wise intersection conjecture

Let $d\geq2$, and let $\mathcal{E}$ be a $\frac12$-sparse antichain of $d$-dimensional dyadic rectangles. Let $\operatorname{sh}(\mathcal{E})$ be the shadow of $\mathcal{E}$. Antichain k-wise intersection conjecture. There exists a finite positive constant $C$, depending only on $d$, such that for every integer $k\geq1$,

$$\sum_{\substack{\mathcal{F}\subseteq\mathcal{E}\\\#(\mathcal{F})=k}}\left|\bigcap_{R\in\mathcal{F}}R\right|\leq C^k(k!)^{d-2}|\operatorname{sh}(\mathcal{E})|.$$

The source states that this geometric estimate is equivalent to the antichain exponential-integrability conjecture. Its two-dimensional specialization is proved in the paper, whereas the all-dimensional statement is left as the conjectural equivalent.