Removing derangements without eliminating all point-mapping derangements

Determine, for n>5, the minimum number of derangements that must be removed from Der(n) so that the remaining set together with the identity contains no transitive subgroup, without removing all derangements that map a given point i to a given distinct point j.

Background

The clique-coclique arguments in the paper connect increased independence numbers with the absence of transitive subgroups in the remaining connection set together with the identity. The authors ask whether this obstruction can be achieved without deleting an entire point-mapping class of derangements.

References

Question 7.4. For n > 5, is there a way to remove derangements from Der(n) so that the resulting set, together with (1), has no transitive subgroups, without removing all the derangements that map i to j?

— A new measure of robustness of Erdős--Ko--Rado Theorems on permutation groups  (2502.14582 - Gunderson et al., 20 Feb 2025) in Question 7.4, Section 7