Pre-Tannakian category for the fourth Delannoy measure μ4

Ascertain whether there exists a pre-Tannakian category associated to the measure μ4 on the oligomorphic group Aut(Q,<), i.e., determine whether the Karoubi envelope of the permutation module category uPerm(G, μ4) admits an abelian envelope.

Background

For the group G=Aut(Q,<), the authors consider four measures μ1–μ4. The classical Delannoy category corresponds to the regular measure μ1, and they establish pre-Tannakian constructions for μ2 and μ3 (the second and third Delannoy categories).

However, for μ4 the existence of an associated pre-Tannakian category remains unresolved.