If one models the electron as a hollow spherical conductor with charge $e$ and radius $a$ then its electrostatic energy is given by:

$$E_{em}=\frac{1}{2}\frac{e^2}{4\pi\epsilon_0a}$$

However if one calculates the momentum in the field of a moving electron then one finds that the total mass in the field is given by:

$$m_{em}=\frac{2}{3}\frac{e^2}{4\pi\epsilon_0c^2a}$$

Therefore we have a discrepancy in energy given by:

$$E_p = \frac{1}{6}\frac{e^2}{4\pi\epsilon_0a}$$

Poincare hypothesised that there must be stresses holding the electron together against the electrostatic repulsion of the charge on its surface. Somehow it must take an energy $E_p$ in order to maintain these Poincare stresses.

Perhaps the electron can be modelled by a conducting spherical shell with a vacuum inside it. Presumably the vacuum would lead to a negative pressure on the charged shell due to the Casimir effect. This pressure must balance the electrostatic repulsion of the charged shell.

According to cosmological models, for example, the vacuum has an equation of state given by:

$$p = -\rho c^2$$

The outward pressure on the surface of the charged sphere due to Coulomb repulsion is given by:

$$p = \frac{1}{2}\epsilon_0\left(\frac{e}{4\pi\epsilon_0a^2}\right)^2$$

This pressure must be balanced by the negative pressure of the vacuum inside the shell. If we substitute into the above equation of state for a vacuum we find that its energy must be given by: $$E_p=\frac{1}{6}\frac{e^2}{4\pi\epsilon_0a}$$

Thus we seem to have accounted for the energy discrepancy between the EM field of a static electron and a moving electron by including the energy of the vacuum inside the electron holding it together.

But in fact the inward pressure on the electron is due to the Casimir effect. This means that it is due to an excess of zero-point electromagnetic modes outside the conducting shell compared to the number of modes inside. Thus the extra energy $E_p$ associated with these extra modes is located outside the shell. This makes sense as we want to account for the discrepancy in the total mass/energy in the field outside the shell.

One can make the following close analogy with the case where one pulls a piston out of a cylinder that is surrounded by normal atmospheric pressure. One has to supply energy to do work against the outside atmosphere. The energy supplied is not stored in the vacuum in the cylinder but rather it is located outside in the surrounding atmosphere.

Does this make sense?