The dimension formula conjecture for self-similar measures

Let $\Phi$ be an iterated function system, let $p$ be a probability vector, and let $\mu$ be the corresponding self-similar measure. The entropy rate is

$$h(\Phi,p)=\lim_{N\to\infty}\frac{1}{N}H(X_1\circ\cdots\circ X_N),$$

where $X_1,X_2,\ldots$ are independent random elements of $\Phi$ distributed according to $p$, and $H$ denotes Shannon entropy. The dimension formula conjecture.

$$\dim \mu=\min\left(1,\frac{h(\Phi,p)}{\sum_{i\in\Lambda}p_i\log|\lambda_i|^{-1}}\right).$$

This folklore conjecture predicts that the dimension is governed by the entropy rate, capped at the ambient dimension. It is motivated by the similarity dimensions obtained from iterated compositions and collapsing exact overlaps; no resolution status is supplied in the source.