Let $\mathcal I$ be a $\sigma$-ideal with Borel base on an uncountable Polish space $X=\bigcup\mathcal I$.

Let $\mathrm{cov}(\mathcal I)$ (resp. $\mathrm{cov}_\sqcup(\mathcal I)$) be the smallest cardinality of a cover of $X$ by (parvise disjoint) Borel sets that belong to the ideal $\mathcal I$.

The cardinal $\mathrm{cov}(\mathcal I)$ is one of four classical cardinal characteristics of an ideal. What about its disjoint modification? Was it considered in the literature?

Observe the following trivial relations between the cardinals $\mathrm{cov}(\mathcal I)$ and $\mathrm{cov}_\sqcup(\mathcal I)$:

1) $\mathrm{cov}(\mathcal I)\le\mathrm{cov}_\sqcup(\mathcal I)$;

2) If $\mathrm{cov}(\mathcal I)\le\omega_1$, then $\mathrm{cov}(\mathcal I)=\mathrm{cov}_\sqcup(\mathcal I)$.

**Problem 1.** Is $\mathrm{cov}(\mathcal I)=\mathrm{cov}_\sqcup(\mathcal I)$?

This problem is especially interesting for the ideal $\mathcal M$ of meager subsets of the real line and for the ideal $\mathcal N$ of Lebesgue null sets in $\mathbb R$.

**Problem 2.** Are the strict inequalities $\mathrm{cov}(\mathcal M)<\mathrm{cov}_\sqcup(\mathcal M)$ and $\mathrm{cov}(\mathcal N)<\mathrm{cov}_\sqcup(\mathcal N)$ consistent?

The negative answer to the second part of problem 2 would give an affirmative answer to Problem 8 of this MO post.