We consider a sequence matching problem involving the optimal alignment score for contiguous sequences; rewarding matches and penalizing for deletions and mismatches. Arratia and Waterman conjectured in [1] that the score constant a(μ, δ) is a strictly monotone function (i) in δ for all positive δ and (ii) in μ if 0 ≤ μ ≤ 2δ. Here we prove that (i) is true for all δ and (ii) is true for some μ.