A remark on the Debs-Saint-Raymond theorem

A theorem of Debs and Saint-Raymond gives sufficient conditions for a $\sigma$-ideal of compact sets to have the covering property. We discuss the necessity of these conditions. Namely, we show that there exists a $\boldsymbol \Pi _{\mathbf{1}}^{\mathbf{1}}$ $\sigma$-ideal that is locally non-Borel, has no Borel basis and has the covering property. This partially answers a question posed by Kechris.