Prove the absence of additional Borel-transform singularities

Prove that the Borel transform of the vacuum-energy perturbation series for the D-dimensional O(N) λ(φ²)² theory has no singularities in the cut z-plane outside the branch cut [S₀,∞), where S₀ is the nearest singularity to the origin.

Background

The conformal mapping used in Eq. (2.6) maps the complex z-plane cut along [S₀,∞) onto the unit disk in the λ-plane. For the mapped perturbative expansion to converge throughout the relevant integration domain, the authors must assume that the Borel transform has no further singularities in the z-plane away from this cut.

The paper explicitly states that a proof of this analytic property is not known. Although the successful Borel summation of correlation functions using the same conformal mapping provides supporting evidence, establishing the absence of additional singularities remains unresolved and is necessary to justify the analytic continuation underlying the resummation method.