Regularity for double phase problems at nearly linear growth

Minima of functionals of the type $$w\mapsto \int_{\Omega}\left[\snr{Dw}\log(1+\snr{Dw})+a(x)\snr{Dw}^{q}\right] \dx\,, \quad 0\leq a(\cdot) \in C^{0, \alpha}\,,$$ with $\Omega \subset \er^n$, have locally H\"older continuous gradient provided $1 < q < 1+\alpha/n$.