Maximum connectivity preserved by quantum isomorphism

Determine the maximum integer k such that every graph that is k-connected remains k-connected under quantum isomorphism.

Background

The paper establishes that k-connectivity is preserved for k at least 2 through its preservation of 2-connectedness and block structure. The authors ask for the strongest universal connectivity-preservation threshold. They note that if Question 6.1 has a positive answer, then this maximum must be finite.

References

What is the maximum k such that k-connectivity is preserved under quantum isomorphism? That is, what is the maximum k such that for every k-connected graph G we have that every graph quantum isomorphic to G is k-connected?

— Block structures of graphs and quantum isomorphism  (2502.19343 - Freslon et al., 26 Feb 2025) in Question 6.3, Section 6, p. 20

Question 6.3. What is the maximum k such that k-connectivity is preserved under quantum isomorphism? That is, what is the maximum k such that for every k-connected graph G we have that every graph quantum isomorphic to G is k-connected?

— Block structures of graphs and quantum isomorphism  (2502.19343 - Freslon et al., 26 Feb 2025) in Question 6.3, Section 6, p. 20