= Solution
Use ideal large plates and neglect edge effects. A Lorentz boost normal to the plates does not contract either transverse plate dimension. Charge is invariant, so charge per plate area remains $\sigma'=\sigma$. <Gauss's law> then gives
$$
\boxed{E_\parallel'=\frac{\sigma'}{\epsilon_0}=\frac\sigma{\epsilon_0}=E_\parallel.}
$$
The plate separation contracts, but the infinite-plate field is <independent> of separation. It is therefore incorrect to infer a stronger field merely from the shorter gap. The ideal field is parallel to the boost, so there is no induced transverse <magnetic field> from this boost.
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