Multiplicative separability of the positive linear functional contrast

Let $A$ be a real-valued positive definite operator representing an attribute, let $M$ represent the action of a mechanism, and let $U$ be a unitary operator diagonalizing $A$. Let $\mathcal{G}$ be the generic subgroup of operators, and define the positive linear functional contrast by

$$C(A)=\langle A,e\rangle,$$

where $e$ is the identity operator. The action is $mA=MAM^*$, where $*$ denotes complex conjugation.

Multiplicative separability conjecture. For all positive definite attributes $A$, whenever $U$ belongs to $\mathcal{G}$, the contrast is multiplicatively separable and

$$\langle C\rangle_{m,x}=C(MM^*)C(A).$$

Such a multiplicative form for the contrast would support the derivation of a forward-backward inequality under one additional assumption. The source does not provide evidence that the conjecture has been resolved.