The Complexity Zoo states that ACC$^0$ is contained in TC$^0$ and links to the paper On ACC and Threshold Circuits. However, what the linked paper proves is that depth-3 threshold circuits of quasipolynomial size can simulate ACC$^0$ circuits.
Wouldn't it be necessary to prove that constant depth threshold circuits of polynomial size simulate them rather than quasipolynomial? Is it obvious how to do the simulation with just a polynomial number of gates?