Characterization of the set of right-bracket regular functions

Determine whether, in the general setting of commuting, trace-preserving actions of unimodular groups on a common semifinite von Neumann algebra, the set of functions $Hbracket{W}{W}$ can be a proper subset of the set of all $rho$-regular functions.

Background

The paper defines $W$ as the set of elements of $L^1(\M)$ that are simultaneously $\alpha$-regular and $\beta$-regular, where $\alpha$ and $\beta$ are commuting, trace-preserving actions of unimodular groups $G$ and $H$. The notation $\Hbracket{W}{W}$ denotes the collection of right-bracket functions formed from pairs of elements of $W$, while $\rho$-regular functions are those in $L^1(H)$ whose right translates span a dense subspace of $L^1(H)$.

The question arises in the context of a Tauberian theorem characterizing when a bounded operator has vanishing-at-infinity bracket functions and belongs to the associated subspace $K$. The theorem uses functions from $\Hbracket{W}{W}$, but the paper does not establish whether every $\rho$-regular function is obtained this way or whether $\Hbracket{W}{W}$ may be strictly smaller.