Small values of the Riemann zeta-function near the critical line

Assuming the Riemann Hypothesis, we study negative moments and small values of the Riemann zeta-function to the right of the critical line. For every fixed $0<k<1/2$, we establish the conjecturally sharp bound $$ \frac1T\int_T^{2T} \Big|ζ(\tfrac12+α+it)\Big|^{-2k}\,dt \ll α^{-k^2}, $$ for $α=o(1