Classification of Hornich–Hlawka Nakano spaces

Prove that if a Nakano space Lp(·)(0, 1), with measurable exponent p(·) : (0, 1) → [1, 2], is Hornich–Hlawka, then p(·) is constant almost everywhere.

Background

The paper constructs several nonconstant exponent functions for which the associated Nakano space is not Hornich–Hlawka, including continuous, smooth, and piecewise-constant examples. These examples motivate the conjecture that essential constancy of the exponent is necessary. Together with the known Hornich–Hlawka property of constant-exponent spaces Lp for 1 ≤ p ≤ 2, the conjecture would yield a complete classification of the Nakano spaces considered.