Regularity for multi-phase problems at nearly linear growth

Minima of the log-multiphase variational integral $$w\mapsto {\int }_{\mathrm{\Omega }}[\mathrm{\mid }Dw\mathrm{\mid }\log (1+\mathrm{\mid }Dw\mathrm{\mid })+a(x)\mathrm{\mid }Dw{\mathrm{\mid }}^q+b(x)\mathrm{\mid }Dw{\mathrm{\mid }}^s]\mathrm{d}x,$$ have locally Hölder continuous gradient under sharp quantitative bounds linking the growth powers $(q,s)$ to the Hölder exponents of the modulating coefficients $a(\cdot )$ and $b(\cdot )$ respectively.