Asymptotic fixed-perimeter Kang-Kim conjecture

Prove that for positive integers 0<m1<m2m and 0<ad, the limit of h(a)(n)-e(m1,m2)(n) as n tends to infinity equals negative infinity when m>2d+2 and positive infinity when m<2d+2.

Background

The authors compare h(a)(n), which counts fixed-perimeter partitions with d-distinct parts at least a, with e(m1,m2)(n), which counts fixed-perimeter partitions whose parts are congruent to m1 or m2 modulo m. This comparison is motivated by Kang and Kim's asymptotic results for ordinary partition functions.

For the fixed-perimeter setting, the authors formulate a conjectured threshold at m=2d+2: they predict that the difference tends to negative infinity below the threshold and positive infinity above it. The critical case is more complicated and is treated only partially by Proposition 1.6.