General comparison with measurable ingredients for integro-differential operators

Assume the symmetry condition and the pointwise ellipticity condition hold. The outcome of the comparison result for measurable ingredients should remain true with the operator $M^+_A$ replaced by $M^+_{CS}$, the appropriate extremal operator for the pointwise ellipticity condition.

General comparison with measurable ingredients conjecture. Under these assumptions, the conclusion of the comparison proposition holds with $M^+_{CS}$ in place of $M^+_A$.

This comparison estimate is expected to hold beyond the previously proved setting and is needed to extend the stochastic homogenization result to more general stationary ergodic integro-differential operators. The source states it as conjectural; no resolution is given here.