Gog and GOGAm pentagons

We consider the problem of finding a bijection between the sets of alternating sign matrices and of totally symmetric self complementary plane partitions, which can be reformulated using Gog and Magog triangles. In a previous work we introduced GOGAm triangles, which are images of Magog triangles by the Schützenberger involution. In this paper we introduce Gog and GOGAm pentagons. We conjecture that they are equienumerated. We provide some numerical evidence as well as an explicit bijection in the case when they have one or two diagonals. We also consider some interesting statistics on Gog and Magog triangles.