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The purpose of the article is to provide partial proofs for two conjecturesgiven by Witte and Forrester in "Moments of the Gaussian $\beta$ Ensembles andthe large $N$ expansion of the densities" with the use of the topologicalrecursion adapted for general $\beta$ Gaussian case. In particular, the paperuses a version at coinciding points that provides a simple proof for some ofthe coefficients involved in the conjecture. Additionally, we propose ageneralized version of the conjectures for all correlation functions evaluatedat coinciding points.

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