Since $\mathbf E = -∇Φ - ∂\mathbf A/∂t$ one expects an oscillating $\mathbf E$ field even in the null of a Hertzian Dipole unless the two right hand side terms cancel -- which they do in the far field of the null.
However, in the near field of the null, the terms do not completely cancel, leaving a residual oscillating E-field.
Since the null has, by definition, no $∇ × \mathbf A$ curl in the oscillating $\mathbf A$, there is no $\mathbf B$ thence no $\mathbf H$ field and therefore no $\mathbf E × \mathbf H$ and since $\mathbf E × \mathbf H$ is the only accepted definition for the dipole's Poynting vector, there is no accepted way for energy to be locally available at points along the dipole's null.
If one places a particle of charge $q$ and mass $m$ along the null, it must experience a force, $\mathbf F=q\mathbf E$ and thence acceleration $\mathbf F=m\mathbf a$.
Where does this energy come from, and how is it delivered without violating locality?