Strong Molino conjecture for singular Riemannian foliations

Let $\mathcal{F}$ be a complete singular Riemannian foliation, and let $\mathscr{C}_{\mathcal{F}}$ be the locally constant sheaf of Lie algebras associated with its structural theorem. Its sections are germs of transverse Killing vector fields. Strong Molino conjecture. The sheaf $\mathscr{C}_{\mathcal{F}}$ is a sheaf of Lie algebras of germs of transverse Killing vector fields. This would extend the regular Molino structure theory to singular Riemannian foliations and would imply Molino's conjecture by making the smooth fields in $\mathscr{C}_{\mathcal{F}}(U)$ transitive on leaf closures. The source does not state a resolution, so the conjecture is recorded as open.