The three-term inequality for weights of free bases of vector-valued modular forms

Let $\rho$ be an irreducible representation, and let $m_k$ denote the multiplicity of forms of even weight $k$ among a free basis for $M(\rho)$ over the ring of scalar-valued modular forms; equivalently, $m_k$ is the multiplicity of $\mathcal{O}(-\frac{1}{2}k)$ in the decomposition of $\mathcal{V}(\rho)$. Three-term inequality. For all $k\in\mathbf{Z}$, one has

$$m_k\leq m_{k+2}+m_{k-2}.$$

This inequality was observed for irreducible $\rho$ in connection with the structure of free bases of vector-valued modular forms. The supplied text does not establish it as a theorem or provide evidence resolving its status, so its conjectural status should be checked against the source.