Quadratic curvature decay is unnecessary for uniqueness on average in surfaces

Let $M$ be the surface setting of the main theorem, so $n=2$, and let the theorem's other hypotheses and notation remain in force. Surface-case conjecture. Theorem 1.1 holds without assuming that the Gaussian curvature decays quadratically. This asks whether the two-dimensional argument extends the main result beyond the stated quadratic Gaussian-curvature decay hypothesis; the source presents it as an open question for future investigation.