Does the $frac{4}{3}$ problem of classical electromagnetism remain in quantum mechanics?

In Volume II Chapter $28$ of the Feymann Lectures on Physics, Feynman discusses the infamous $\frac43$ problem of classical electromagnetism. Suppose you have a charged particle of radius $a$ and charge $q$ (uniformly distributed on the surface). If you integrate the energy density of the electromagnetic field over all space outside the particle, you'll get the total electromagnetic energy, which is an expression proportional to $c^2$. The energy divided by $c^2$ is what we usually call the mass, so if we calculate the "electromagnetic mass" in this manner we'll get $m = \frac{1}{2}\frac{1}{4\pi\epsilon_0}\frac{q^2}{ac^2}$. If, on the other hand, you took the momentum density of the electromagnetic field and integrated it over all space outside the particle, you'd get the total electromagnetic momentum, which turns out (for $v<) to be proportional to the velocity of the particle. The constant of proportionality of momentum and velocity is what we call mass, so if we calculated the electromagnetic mass in this way we would get $m = \frac{2}{3}\frac{1}{4\pi\epsilon_0}\frac{q^2}{ac^2}$, which is $\frac{4}{3}$ times the value we got before! That is the $\frac43$ problem.

Feynman claims that this fundamental issue remains when we move to quantum electrodynamics. Was he right, and if so has the situation changed since the $1960's$ when he was writing? I've seen claims on the Internet (I don't have the links) that the $\frac43$ problem is still there in QED, but instead of $\frac{4}{3}$ the coefficient is something closer to $1.$ Is that true, and if so what's the coefficient? All of this is of course related to issues of self-energy and renormalization.

Any help would be greatly appreciated.