An HKᵣ-Integrable Function Which is Pₛ-Integrable for no s

Given arbitrary r≥ 1 r ≥ 1 , we construct an HK ᵣ r -integrable function which is not P ₁ 1 -integrable. This is an extension of a recently published construction [Musial, P., Skvortsov, V., Tulone, F.: The HK ᵣ r -integral is not contained in the P ᵣ r -integral. Proceedings of the American Mathematical Society 150 (5), 2107–2114 (2022)].