Well-balanced classes are asymptotically negligible for repetitive finite-local-complexity tilings

Let ${\bf T}$ be a tiling that is repetitive and has finite local complexity, and let $\Omega_{\bf T}$ be its continuous hull. Consider the top-degree real cohomology group $H^n(\Omega_{\bf T},\mathbb R)$, where $n$ is the dimension of the tiling. A cohomology class is well-balanced if it has the corresponding bounded mass-transport property, and it is asymptotically negligible if it has the corresponding weakly pattern-equivariant transport property.

Well-balanced-class conjecture. Every well-balanced class in

$$H^n(\Omega_{\bf T},\mathbb R)$$

is asymptotically negligible.

The claim would identify the two cohomological conditions governing bounded and weakly pattern-equivariant mass transport for every repetitive tiling with finite local complexity. The supplied text does not state whether this conjecture has been proved or disproved.