Characteristic morphisms for $S/\!/\eta,\sigma$, $S/\!/\nu,\sigma$, and $S/\!/\sigma$

Consider the commutative $S$-algebras and maps displayed below, with the maps composed with the standard maps indicated in the source:

$$S/\!/\eta,\sigma\longrightarrow MU\longrightarrow kU\longrightarrow H\mathbb{Z}_{(2)},$$ $$S/\!/\nu,\sigma\longrightarrow MSp\longrightarrow kO,$$

and

$$S/\!/\sigma\longrightarrow tmf.$$

A map is a characteristic morphism when its source is a characteristic for its target.

Characteristic morphism conjecture. Each of the three displayed maps is a characteristic morphism.

The source has already constructed the relevant maps and notes that $S/\!/\sigma\to tmf$ is expected to be characteristic. The subsequent discussion reformulates the three assertions as equalities of kernels on stable homotopy groups; their resolution is not supplied here.