For the sake of documentation, I introduce here a proof I found today,
[A note on “Measuring propagation speed of Coulomb fields” by R. de Sangro, G. Finocchiaro, P. Patteri, M. Piccolo, G. Pizzella][1]

Basically they followed Feynman's suggestion by deriving the electrical field from the Heaviside-Feynman formula and comparing the result to standard result obtained from the Lienerd-Wiechert potentials found in many textbooks.

  [1]: https://link.springer.com/article/10.1140/epjc/s10052-016-4108-7