Modular collapse conjecture for locally compact Moufang-type quasigroups

Let $Q$ be a locally compact Hausdorff topological quasigroup satisfying the Moufang-type identity $(N1)$. Assume that $Q$ admits a quasi-invariant measure with modular cocycle $j$.

Modular collapse conjecture. The cocycle collapses:

$$j\equiv 1.$$

In particular, the translation geometry becomes unimodular in the measure-theoretic sense. The conjecture proposes a measure-theoretic interpretation of Kunen's theorem by asserting that the $(N1)$ identity forces unimodularity whenever a suitable quasi-invariant measure exists. The paper develops the cocycle framework but does not establish this collapse.