Is the following generalised compactness property of Borel sets in a Polish space consistent with ZFC?


($\*$) Let $\mathcal{B}$ be a family of $\aleph\_1$-many Borel sets. If $\bigcap \mathcal{B} = \emptyset$, then, for some countable $\mathcal{C} \subseteq \mathcal{B}$, it holds that $\bigcap \mathcal{C} = \emptyset$.


In the special case that $\mathcal{B}$ is a family of open sets, it is not hard to show that property ($\*$) is a consequence of $\neg\mathrm{CH} + \mathrm{MA}$ (where $\mathrm{CH}$ is the Continuum Hypothesis and $\mathrm{MA}$ is Martin's Axiom)—more specifically, a consequence of $\mathrm{MA}\_{\aleph\_1}$. Also, trivially, this special case of ($\*$) itself implies $\neg \mathrm{CH}$.


However, I am stuck as to the consistency of ($\*$) for general Borel sets. Actually, I have been unable to show consistency even when $\mathcal{B}$ is restricted to families of $F\_\sigma$ sets.



A compactness property for Borel sets
It is worth noting that a construction provided by Hausdorff more than a hundred years before the result of Kubis and Vejnar also provides a counterexample. A Hausdorff gap is a family of subsets of the natural numbers $A\_\xi, B\_\xi$ for $\xi\in\omega\_1$ such that 
$A\_\xi \subseteq^\* A\_\eta \subseteq^\* B\_\eta \subseteq^\* B\_\xi$ for $\xi< \eta$ but such that there is no subset of $\mathbb N$ such that $ A\_\xi \subseteq^\* X \subseteq^\* B\_\xi$ for all $\xi$. (Here $\subseteq^\*$ means inclusion except for a finite set.) Letting $S\_\xi$ be the Borel set of all $X\subseteq \mathbb N$ such that $ A\_\xi \subseteq^\* X \subseteq^\* B\_\xi$ yields the counterexample.


