Maximum degree of 4-connected claw-free 1-planar graphs

Prove or disprove that every 4-connected claw-free 1-planar graph has maximum degree at most 8.

Background

The paper proves that every claw-free 1-planar graph has maximum degree at most 10 and that every 6-connected claw-free 1-planar graph has maximum degree at most 8. The authors construct 3-connected examples attaining maximum degree 10 but state that they have not found a 4-connected example with maximum degree 10. They therefore conjecture that the maximum-degree bound can already be reduced to 8 under 4-connectivity, and they note that this bound would be tight because a corresponding 4-connected example with maximum degree 8 is constructed.

References

However, we have not yet found a 4-connected claw-free 1-planar graph with maximum degree 10. We even believe that, in this case, the bound 10 could be improved. That is, the 6-connectedness condition in Theorem 1.4 may be reduced to 4-connectedness. Conjecture 4.1. Let G be a claw-free 1-planar graph. If G is 4-connected, then ∆(G) ≤ 8.

— On the properties of claw-free 1-planar graphs  (2501.15124 - Zhang et al., 25 Jan 2025) in Concluding remarks and open problems, Conjecture 4.1, page 10