Relative quantumness data-processing conjecture for CoP channels

Let $\mathcal H$ be a finite-dimensional Hilbert space, let $\rho,\sigma\in\mathcal D(\mathcal H)$ satisfy $[\rho,\sigma]\neq0$, and let $\rho_\varepsilon,\sigma_\varepsilon$ be their regularized states. A commutativity-preserving (CoP) channel is a CPTP map $\mathcal N:\mathcal B(\mathcal H)\to\mathcal B(\mathcal H)$ that maps every commuting pair of states to a commuting pair. Assume

$$\operatorname{supp}\!\bigl(\mathcal N(\rho_\varepsilon)\bigr)\subseteq\operatorname{supp}\!\bigl(\mathcal N(\sigma_\varepsilon)\bigr).$$

Relative quantumness data-processing conjecture. Under these assumptions,

$$Q(\rho_\varepsilon\|\sigma_\varepsilon)\geq Q\left(\mathcal N(\rho_\varepsilon)\|\mathcal N(\sigma_\varepsilon)\right)\geq0.$$

The conjecture asks whether relative quantumness is monotone under physically relevant commutativity-preserving channels for non-commuting inputs. The support condition ensures that the regularized functional remains well-defined even when channel outputs are rank-deficient; the data-processing inequality is not established in the supplied text.