The RBC & UKQCD collaborations have successfully employed G-parity boundary conditions in the measurement of $K\to\pi\pi$ decays to obtain a physical decay with the two-pion ground state, at the cost of a significant increase in computational expense. We report on new theoretical/algorithmic developments based upon the properties of the Dirac operator under complex conjugation that have been exploited to achieve highly significant computational cost reductions, and explain their impact on our ongoing effort to repeat the calculation on finer lattices.