Monotonicity conjecture for sets of quantum p-cost operators

For a separable Hilbert space $\mathcal{H}$, let $\mathbf{C}_p(\mathcal{H})$ denote the set of all possible $p$-cost operators. Monotonicity conjecture for quantum $p$-cost operators. For parameter values $0<p\leq p'\leq\infty$, one should have

$$\mathbf{C}_{p}(\mathcal{H})\subseteq \mathbf{C}_{p'}(\mathcal{H}).$$

The conjecture formalizes the expectation that increasing the parameter enlarges the set of possible cost operators. No resolution is given in the source.