Bell–Ho–Strichartz conjecture on the asymptotic Kusuoka-measure distribution

Let $K$ be the Sierpiński gasket, let $W_m$ denote the set of words of length $m$, and let $c^{(w)}\break$ be the vector of one-step Kusuoka-measure distribution ratios associated with $w\break\breakin W_*$. Let $r$ and $\theta$ be the polar-coordinate maps on the disk $\mathbb{D}$ centered at $^t(1/3,1/3,1/3)$, and let $\lambda_m$ be the uniform probability distribution on $W_m$.

Bell–Ho–Strichartz conjecture. As $m\to\infty$, the law of $r\circ c^{(w)}$ under $\lambda_m$ converges to the Dirac measure at $\sqrt{8/75}$, while the law of $\theta\circ c^{(w)}$ under $\lambda_m$ converges to an absolutely continuous measure on $(-\pi,\pi]$.

The conjecture describes the limiting radial and angular distribution of the Kusuoka-measure ratios on the Sierpiński gasket. Bell, Ho, and Strichartz posed it in connection with their result that all such vectors lie in $\mathbb{D}$ and can approach its boundary; the supplied text gives no evidence that the conjecture has been resolved.