An HKᵣ-integrable function which is Pₛ-integrable for no s

Given arbitrary r≥1, we construct an HKᵣ-integrable function which is not P₁-integrable. This is an extension of Musial et al.\ construction published recently in [Musial, P., Skvortsov, V., Tulone, F.: The HKᵣ-integral is not contained in the Pᵣ-integral. Proc. Amer. Math. Soc. {150}(5), 2107--2114 (2022)].