The third author's rigidity conjecture for order-type limits

Let $cmu$ be a measure on $cmathbf{R}^2$ that charges no line, and suppose that its support has Hausdorff dimension strictly greater than $1$. Let $cell_cmu$ be the order-type limit realized by $cmu$. The third author's rigidity conjecture. Every measure that realizes $cell_cmu$ is projectively equivalent to $cmu$.

This conjecture weakens the theorem's hypotheses from compact support and support with non-empty interior to the condition that the support has Hausdorff dimension greater than $1$. It predicts uniqueness of the realizing measure up to projective equivalence, but the supplied text gives no resolution.