We show that if a cusped Borel Anosov representation from a lattice Γ ⊂ PGL₂(\,R\,) Γ ⊂ PGL 2 ( R ) to PGLd(\,R\,) PGL d ( R ) contains a unipotent element with a single Jordan block in its image, then it is necessarily a (cusped) Hitchin representation. We also show that the amalgamation of a Hitchin representation with a cusped Borel Anosov representation that is not Hitchin is never cusped Borel Anosov.
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Lee et al. (2024) studied this question.
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