Reduced integrability conjecture for diffusion-invariant stochastic dynamical systems

Let $X = X_0 + \sum X_i \circ \dfrac{dB_t^i}{dt}$ be an SDS on a manifold $M$. Suppose that $X$ is integrable and diffusion invariant with respect to an action of a compact Lie group $G$ on $M$. The quotient has a regular part denoted by the regular part of $M/G$, and the reduced system of $X$ is defined there. Reduced integrability conjecture. The reduced system of $X$ on the regular part of the quotient space $M/G$ is also an integrable SDS. This conjecture concerns whether integrability is preserved under reduction by a compact symmetry group; the supplied source does not state whether it has been proved or disproved.