The singularity conjecture for the weighted-cluster generating function

Let $\Gamma'$ be a decoration of a $d$-regular tree, and let $g(x)$ be the generating function for weighted clusters containing the origin. Let $D$ denote the exponential growth factor arising in the associated cluster-counting construction. Generating-function singularity conjecture. The closest singularity of $g(x)$ to the origin is at $x=\frac{1}{D}$, and near $\frac{1}{D}$ its dominant term is $(1-Dx)^{\frac{1}{2}}$. This conjectured square-root singularity would imply a critical exponent of $\frac{3}{2}$ for the decorated graph. The paper does not prove the assertion in the general case.