Borel summability of the loop vertex expansion in non-integer dimension

Let $D$ be real with $0\leq D<2$, and let $A_{D,\bar{\mathcal{T}}}$ denote the $D$-dimensional tree amplitude obtained by summing the Feynman amplitudes over graphs containing the tree-like structure $\bar{\mathcal{T}}$. For sufficiently small coupling $\lambda$, Borel-summability conjecture.

$$\sum_{\bar{\mathcal{T}}}\left\lvert A_{D,\bar{\mathcal{T}}}\right\rvert<\infty$$

and the result is the Borel sum of the initial perturbative series. The authors motivate this conjecture by the known convergence of the loop vertex expansion for $D=0$ and $D=1$; the proposed range $0\leq D<2$ is the regime in which no ultraviolet divergences require renormalization.