Characterization of D-separable compact spaces and equality of cardinal invariants

Let $X$ be a compact space. A

$$-disjoint$$

-base is a

-base that is a countable union of pairwise disjoint collections. Let $d\delta(X)$ be the least cardinal

such that every dense subset of $X$ is the union of

many discrete sets, and let $d\pi(X)$ be the least cardinal

such that $X$ has a

$$-base that is the union of$$

many disjoint collections.

Characterization and cardinal-invariant conjecture. \begin{enumerate} \item A compact space $X$ is D-separable if and only if $X$ has a $\sigma$-disjoint $\pi$-base. \item For every compact space $X$, $d\delta(X)=d\pi(X)$. \end{enumerate}

These problems ask for a characterization of D-separability for compact spaces and for an equality between two cardinal invariants associated with discrete decompositions of dense sets and $\pi$-bases. The supplied text presents them as open problems and gives no resolution.