Complex-dimension conjecture for non-interval fractals

Let $K$ be the fractal under consideration, and assume the conditions of Theorems~ and~. A complex dimension is called nontrivial when it is one of the complex dimensions considered in the cited fractal zeta-function framework other than the trivial dimensions. Complex-dimension conjecture. There are nontrivial complex dimensions if and only if $K$ is not homeomorphic to an interval. This predicts that non-interval topology is exactly the obstruction to having only trivial complex dimensions, under the stated hypotheses. The supplied text gives no resolution of this conjecture.