 Solution

The matter-like scaling is special to a quadratic minimum. More generally, the effective equation of state of an oscillating scalar field in $V\propto |\varphi {|}^n$ isFor example, a quartic minimum has $⟨w⟩=1/3$ and redshifts like radiation in cosmology. Potentials without a stable minimum, oscillations whose period is not short relative to the Hubble time, and significant decay of the inflaton into other particles also invalidate the matter-like argument.