Inversions and the Gog-Magog problem

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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 left Gog and GOGAm trapezoids. We conjecture that they are equienumerated, and we give an explicit bijection between such trapezoids with one or two diagonals. We also study the distribution of inversions and coinversions in Gog triangles.