Algebraic-integrability or logarithmic-component conjecture at second degree defect one

Let $\Sigma\subset\textsf{Fol}_d(\mathbb P^n_{\mathbb C})$ be an irreducible component, and let $\delta_2(\Sigma)$ denote its second subdistribution invariant. For a foliation $\mathcal F\in\Sigma$, algebraic integrability means that all leaves of $\mathcal F$ are algebraic. Let $\textsf{Log}_{(d_1,\ldots,d_r)}(\mathbb P^n_{\mathbb C})$ denote the logarithmic component associated with integers $d_1,\ldots,d_r$.

Algebraic-integrability or logarithmic-component conjecture. If

$$\delta_2(\Sigma)=\deg(\mathcal F)-1,$$

then either every foliation $\mathcal F\in\Sigma$ is algebraically integrable, or there exist integers $d_1,\ldots,d_r$ such that

$$\Sigma=\textsf{Log}_{(d_1,\ldots,d_r)}(\mathbb P^n_{\mathbb C}).$$

The claim gives a dichotomy between algebraic integrability and membership in a logarithmic component. The surrounding text contains related results, but the supplied material does not establish this exact component-level statement as resolved.