electromagnetism - How do I derive the Lorenz gauge from the continuity equation?

I was reading my old electromagnetics book (Elements of Electromagnetics, by Sadiku, 3rd edition) and after the author explained what the Lorenz gauge is mathematically and why it is useful in decoupling the scalar electrical potential from the vector magnetic potential to produce the wave equations, the author made a side comment below eq. (9.52) that the Lorenz gauge $$\nabla \cdot \vec A = -\mu\varepsilon \dfrac{\partial V}{\partial t}$$ can be derived from the continuity equation.

I don't have a background in relativity. I am an electrical engineering student. So given these limitations, how would I go about deriving the Lorenz gauge from the continuity equation?

Note: I am sticking to the conventions of my book in writing equations.