Let us consider two completely positive and trace preserving (CPTP) linear maps $\Phi:M_d(\mathbb{C})\rightarrow M_{d_1}(\mathbb{C})$ and $\Phi_c:M_d(\mathbb{C})\rightarrow M_{d_2}(\mathbb{C})$ that are complementary to each other, i.e., there exists an isometry $V:\mathbb{C}^d\rightarrow \mathbb{C}^{d_1}\otimes \mathbb{C}^{d_2}$ such that
$$\forall X\in M_d(\mathbb{C}): \quad \Phi(X) = \text{Tr}_2 (VXV^\dagger) \quad\text{and}\quad \Phi_c(X) = \text{Tr}_1 (VXV^\dagger),$$
where $\text{Tr}_1$ and $\text{Tr}_2$ denote partial traces over $M_{d_1}(\mathbb{C})$ and $M_{d_2}(\mathbb{C})$, respectively. We define the adjoints $\Phi^*: M_{d_1}(\mathbb{C})\rightarrow M_{d}(\mathbb{C})$ and $\Phi_c^*:M_{d_2}(\mathbb{C})\rightarrow M_d(\mathbb{C})$ of these maps uniquely by the following relations: $\forall X\in M_d(\mathbb{C}),\, \forall Y\in M_{d_1}(\mathbb{C}),\, \forall Z\in M_{d_2}(\mathbb{C}):$
$$ \text{Tr}[\Phi(X)Y]=\text{Tr}[X\Phi^*(Y)] \quad\text{and}\quad \text{Tr}[\Phi_c(X)Z]=\text{Tr}[X\Phi_c^*(Z)].$$
It is well-known that the adjoints of CPTP maps are unital (i.e., they map the identity matrix to the identity matrix) and completely positive. I am interested in the following composed maps:
$$\Phi^*\circ\Phi: M_d(\mathbb{C})\rightarrow M_d(\mathbb{C}) \quad\text{and}\quad \Phi_c^*\circ\Phi_c: M_d(\mathbb{C})\rightarrow M_d(\mathbb{C}),$$
which are again guaranteed to be completely positive (notice that these maps are neither trace-preserving nor unital in general). If people have looked at these kinds of compositions before, I would love to get hold of a reference.
In particular, I want to study the supports (or ranges) of the images of rank one projectors under the above maps. If the map $\Phi$ is also completely copositive, i.e. $\Phi\circ T$ is again a CPTP map and hence $T\circ \Phi^*$ is again unital and completely positive (where $T:M_d\rightarrow M_d$ is the transpose map), I claim that for all unit vectors $v\in\mathbb{C}^d$, the following inclusion holds $$ \text{supp}\,\, [\Phi_c^* \circ \Phi_c](vv^*) \subseteq \text{supp}\,\, [\Phi^* \circ \Phi](vv^*).$$
Any help in proving/disproving the above claim would be greatly appreciated. Thanks!