Must the photon mass be zero?

The old query concerning longitudinal waves, which already beset the elastic theory of light, has in our day revived in the form expressed in the title. Maxwell's field laws are a singular limiting case in that they admit but transversal waves. If this held only in however close an approximation, some fundamental laws of radiation would seem to be affected by a factor $\frac{3}{2}$, on account of the 'third degree of freedom'. If so, this would render even Maxwell's theory suspect, for we are loath to accept as an adequate description of nature a limiting case whose predictions differ grossly and discontinuously from those reached by a sufficiently close approach to the limit. We show here in the simple, if fictitious, example of an ideal conductor, that by extending Proca's field equations in a plausible fashion to the interior of matter the discontinuity is avoided and the correct factors (not $\frac{3}{2}$ thereof) are already reached with a rest-mass at the upper limit, imposed anyhow by other well-known considerations.