Vanishing conjecture for LIP cohomology of almost basic sets

An almost basic set is a set of the type defined in the paper, and $H^k(U,\mathcal{LIP})$ denotes its degree-$k$ cohomology with coefficients in the sheaf of locally integer-valued piecewise functions. Vanishing conjecture. For every almost basic set $U$ and every integer $k>0,$

$$H^k(U,\mathcal{LIP})=0.$$

The conjecture extends the preceding vanishing theorem for almost star-like sets and would establish triviality of all positive-degree cohomology groups for the broader class of almost basic sets. Its status is not determined by the supplied text.