Unimodality of partitions in near-rectangular Ferrers diagrams

We look at the rank generating function Gλ of partitions inside the Ferrers diagram of some partition λ, investigated by Stanton in 1990, as well as a closely related problem investigated by Stanley and Zanello in 2013. We show that Gλ is not unimodal for a larger class of 4-part partitions than previously known, and also that if the ratios of parts of λ are close enough to 1 (depending on how many parts λ has), or if the first part is at least half the size of λ, then Gλ is unimodal.