The $\psi(3686)$ is identified as the radial excitation of the $J/\psi$. Based on perturbative QCD, the branching ratio of the $\psi(3686)$ into some final hadron state should be approximately 13% of the branching ratio of the $J/\psi$ to that same hadron final state. This is called the "13% rule". However, certain decay channels such as the $\rho\pi$ severely violate this 13\% rule. Using the extended Linear Sigma Model, we study the effect a small mixing angle between the $\psi(3686)$ and the vector glueball can have on the 13% rule. We show that in a simple model the mixing can already suppress $\psi(3686)$ decays sufficiently to match the $\rho\pi$ observation, but to fully describe the data a more sophisticated model is necessary. We also show that matching to data can tell us about the magnitude of glueball decay widths.