Cofibrancy preservation conjecture for equivariant colored categories and operads

Let $\mathfrak{C} \to \mathfrak{D}$ be an injection of colors. Write $\mathsf{Cat}_{\mathfrak{D},{\mathcal F}}^G({\mathcal V})$ and $\mathsf{Op}_{\mathfrak{D},{\mathcal F}}^G({\mathcal V})$ for the model categories of $G$-equivariant $\mathfrak{D}$-colored categories and operads, respectively, and similarly for $\mathfrak{C}$. The injection induces pullback functors

$$\mathsf{Cat}_{\mathfrak{D},{\mathcal F}}^G({\mathcal V}) \xrightarrow{\varphi^{**}} \mathsf{Cat}_{\mathfrak{C},{\mathcal F}}^G({\mathcal V}), \qquad \mathsf{Op}_{\mathfrak{D},{\mathcal F}}^G({\mathcal V}) \xrightarrow{\varphi^{**}} \mathsf{Op}_{\mathfrak{C},{\mathcal F}}^G({\mathcal V}).$$

Cofibrancy preservation conjecture. These pullback functors preserve cofibrations between cofibrant objects. This generalizes the Interval Cofibrancy Theorem, which is the special case of the inclusion $\{0\} \to \{0,1\}$ for the category side. Establishing the conjecture would provide a uniform cofibrancy result for restriction along injections of color sets in equivariant colored categories and operads.