Poissonization functor conjecture for Dirac structures

Let $E$) be a Courant algebroid and let $L\subset E$ be a Dirac structure admitting a Lagrangian complement $L'\subset E$. A Poissonization functor for Dirac structures assigns a spanning Poisson structure $(L',p^L)$ to the Lagrangian complement, and the datum of $L$ as a Lie algebroid with an isotropic embedding into $E$ is equivalent to the spanning Poisson structure $(L',p^L)$. This would generalize the correspondence between Lie algebroids and linear Poisson structures and, when available in suitable generality, yield a functorial assignment preserving products and sending Dirac morphisms to coisotropic relations.