Elements of proof for conjectures of Witte and Forrester about the combinatorial structure of Gaussian Beta Ensembles

The purpose of the article is to provide partial proofs for two conjectures given by Witte and Forrester in Moments of the Gaussian $beta$ Ensembles and the large $N$ expansion of the densities with the use of the topological recursion adapted for general $beta$ Gaussian case. In particular, the paper uses a version at coinciding points that provides a simple proof for some of the coefficients involved in the conjecture. Additionally, we propose a generalized version of the conjectures for all correlation functions evaluated at coinciding points.