Well-definedness conjecture for the Sinh-Gordon bootstrap correlation functions

Let $k\in\mathbb{N}$, let $G\in\mathcal{S}\big((\mathbb{R}^{1,1})^k\big)$, and let $\alpha_1,\dots,\alpha_k$ be labels of solutions to the system a)–d). For each $\boldsymbol{n}\in\mathbb{N}^{\frac{k(k-1)}{2}}$, let $\mathcal{I}_{\boldsymbol{\alpha}_k}^{(\boldsymbol{n})}[G]$ denote the corresponding contribution, and write $\boldsymbol{\alpha}_k=(\alpha_1,\dots,\alpha_k)$. Well-definedness conjecture. The $k$-point correlation function

$$\mathcal{W}_{\boldsymbol{\alpha}_k}[G]=\sum_{\boldsymbol{n}\in\mathbb{N}^{\frac{k(k-1)}{2}}}\mathcal{I}_{\boldsymbol{\alpha}_k}^{(\boldsymbol{n})}[G]$$

is given by an absolutely convergent series and defines a tempered distribution. The preceding truncated sums are finite and hence define tempered distributions; convergence of the full series is known for two-point functions with mutually purely space-like support, while the general convergence problem remains open. Establishing this conjecture would provide the well-defined correlation functions needed for the bootstrap construction to satisfy the Wightman axioms.