Endpoint Hornich–Hlawka property of ℓp0(3)

Determine whether the three-dimensional sequence space ℓp0(3), with p0 = log 3/(log 3 − log 2), is Hornich–Hlawka.

Background

The paper reduces the classification of ℓp(N) in the range 2 < p ≤ p0 to the endpoint space ℓp0(3): if this endpoint is Hornich–Hlawka, then all ℓp(N) with N ≥ 3 and 2 < p ≤ p0 are Hornich–Hlawka. The conjecture is motivated by numerical exploration that did not produce a counterexample, but the endpoint property remains unresolved in the paper.