Gradient higher integrability of bounded solutions to parabolic double-phase systems

We prove that bounded solutions to degenerate parabolic double-phase problem modelled upon $$u_t-\dv(|\na u|^{p-2}\na u+a(x,t)|\na u|^{q-2}\na u)=-\dv(|F|^{p-2}F+a(x,t)|F|^{q-2}F)\,,$$ where a nonnegative weight $a$ is $\alpha$-H\"older continuous in space and $\tfrac \alpha 2$-H\"older continuous in time, have locally higher integrable gradients for the sharp range of exponents $p<q\le p+\alpha$.