The orthogonal-complement conjecture for transitive subspaces of an operator and its adjoint

Let $T$ be an operator, let $T^*$ be its adjoint, and let $d4dc_1$ and $d4dc_2$ be subspaces. Assume that $T$ is $d4dc_1$-transitive and $T^*$ is $d4dc_2$-transitive. Orthogonal-complement conjecture. Then

$$\mathcal {M}_2=\mathcal {M}_1^\perp.$$

The question concerns the relation between the subspaces associated with an operator and its adjoint in the setting of subspace-hypercyclic dynamics; the supplied text does not indicate whether this assertion has been proved or disproved.