Hajnal–Juhász basis conjecture for regular Hausdorff spaces

Let $X$ be a regular Hausdorff space. A space is hereditarily separable if every subspace is separable, and hereditarily Lindelöf if every subspace is Lindelöf. Hajnal–Juhász basis conjecture. The following are equivalent:

1. $X$ is hereditarily separable.
2. $X$ is hereditarily Lindelöf.
3. $X$ does not contain an uncountable discrete subspace.

This is a proposed equivalence between standard hereditary separation and covering properties; the supplied material gives no resolution.