Dirac reduction conjecture for spanning Poisson structures

Let $E$ be a Courant algebroid with a pair of transversal Dirac structures $L,L'\subset E$. Assume that $E$ reduces to $E_r$ and that $L$ and $L'$ reduce to Dirac structures $L_r,L'_r\subset E_r$. Dirac reduction conjecture. The spanning Poisson structure $(L',p^L)$ reduces to the spanning Poisson structure $(L'_r,p^{L_r})$. This is proposed as the Dirac analogue of the stated linear Poisson and presymplectic reduction theorems, and would establish compatibility of the Poissonization assignment with reduction.