A constrained optimization problem for the Fourier transform: Quantitative analysis

Among functions $f$ majorized by indicator functions $1_E$, which functions have maximal ratio $\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_q\mathrm{/}\mathrm{\mid }E{\mathrm{\mid }}^{1\mathrm{/}p}$? We establish a quantitative answer to this question for exponents $q$ sufficiently close to even integers, building on previous work proving the existence of such maximizers.