Cascini–Spicer's algebraic integrability conjecture for foliated triples

Let $(X,\mathcal{F},B)$ be a $\mathbb{Q}$-factorial projective foliated triple, such that $\mathcal{F}$ is algebraically integrable, $B$ is a $\mathbb{Q}$-divisor, $(X,B)$ is klt, and one of the following cases holds: $(X,\mathcal{F},B)$ is F-dlt, or $(X,\mathcal{F},B)$ is canonical. Cascini–Spicer's conjecture. Then there exists a morphism

$$f:X\rightarrow Y$$

which induces $\mathcal{F}$. This concerns whether algebraically integrable foliations under these singularity assumptions arise from morphisms; the supplied text gives no resolution status.