Bijection between positive odd-rank partitions and two-color partitions

Construct a bijection between the partitions of a natural number with positive odd rank and the two-color partitions in which the smallest part is an even number 2n and all red parts are even and lie in the interval (2n, 4n].

Background

Theorem 1.2 establishes, by analytic methods, that the number of partitions of n with positive odd rank equals the number of two-color partitions whose smallest part is even, say 2n, and whose red parts are even and lie strictly between 2n and 4n. The equality is currently proved through generating functions rather than by a direct combinatorial correspondence.

The authors explicitly ask whether a bijection can be constructed between these two partition sets, which would provide a combinatorial explanation for the identity.