Uniqueness conjecture II for the length vector of AF$_\beta$ gravitational instantons

In the setting of Theorem, let $\bm{l}=\bm{l}(\beta)$ be the length vector associated with the construction of $\operatorname{AF_\beta}$ gravitational instantons. Uniqueness conjecture, II. For all $\beta\in(0,1)$, the length vector $\bm{l}=\bm{l}(\beta)$ is unique. This conjecture concerns the uniqueness question for the proposed gluing construction of $\operatorname{AF_\beta}$ gravitational instantons for special ranges of $\beta$; the source does not provide a resolution.