Monotonicity of proportional fair capacity for Gaussian broadcast channels

Let $C_{PF}(\mathbf{N},P_T)$ denote the proportional fair capacity of a Gaussian broadcast channel with noise vector $\mathbf{N}$ and total power $P_T$. Let $\preceq$ denote the componentwise ordering of noise vectors. Monotonicity conjecture. If $\mathbf{N}\preceq\mathbf{N}'$, then

$$C_{PF}(\mathbf{N},P_T)\leq C_{PF}(\mathbf{N}',P_T).$$

This conjecture was motivated by numerical examples; the supplied text does not establish it in general, so its resolution remains open.