Near-mirror realisation of low asymmetric depth

Prove that every asymmetric IPR fullerene with asymmetric depth $2$ or $3$ has its depth realised by a near-mirror partial automorphism.

Background

Computationally, every one of the 727 asymmetric IPR fullerenes of depth 2 or 3 with at most 118 vertices has a depth-realising partial automorphism that is self-inverse and reverses the cyclic order at every interior vertex. The paper formalises this property as being near-mirror and states it as a conjecture for all asymmetric IPR fullerenes in the low-depth regime.

References

Let $F$ be an asymmetric IPR fullerene with $d(F) \in {2,3}$. Then $d(F)$ is realised by a near-mirror partial automorphism.

— Extremal Asymmetric Depth of Planar Graphs and Hidden Near-Mirror Symmetries of IPR Fullerenes  (2609.02585 - Pastorek, 2 Sep 2026) in Conjecture 1, Section 7, “Concluding remarks”