Molecules of an affine FPF $W$-graph and an asymptotic row-Beissinger correspondence

Kazhdan--Lusztig $W$-graphs encode the cell structure of Hecke algebras, while their bidirected connected components are called molecules. In finite type~$A$, the Robinson--Schensted correspondence describes cells and molecules, and Beissinger's row insertion constructs the common tableau associated with an involution. In affine type~$A$, the affine matrix-ball construction assigns an affine permutation a pair of tabloids together with a dominant weight, and Marberg introduced affine FPF $W$-graphs indexed by affine fixed-point-free involutions. We prove that, for an affine fixed-point-free involution, complete two-cycle truncations followed by finite row Beissinger insertion asymptotically recover both its common AMBC tabloid and its dominant weight. We also identify the bidirected edges of $\Gamma_n^{\m}$ with dual equivalence moves under AMBC, obtaining a classification of its molecule.