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Instanton L-space property for satellite pattern and companion

Prove that if K=P(C) is an instanton L-space knot in S^3, then both the pattern knot P(U) in S^1×D^2 and the companion knot C in S^3 are instanton L-space knots.

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Background

The paper shows that for an instanton L-space satellite knot K=P(C), the pattern P(U) and the companion C are fibered (Lemma 2.1), paralleling known Heegaard Floer results. However, the stronger statement that P(U) and C are instanton L-space knots is not yet established.

A Heegaard Floer analogue is known (Hanselman–Rasmussen–Watson), but the instanton Floer version remains open, reflecting a gap between these Floer theories in the satellite setting.

References

It should be true that if K = P(C) is an instanton L-space knot then P(U) and C are as well. The Heegaard Floer analogue of this claim is a theorem of Hanselman, Rasmussen, and Watson [HRW], but the instanton version remains open.

— Torus knots, the A-polynomial, and SL(2,C) (2405.19197 - Baldwin et al., 29 May 2024) in Remark following Lemma 2.1 (Preliminaries, Subsection on Instanton L-space knots)