Generating series for Laplace transforms of Hochschild-arbor volume functions

Derive and prove the conjectural generating-series identity for the Laplace transforms of the volume functions of the polytopes Q_{t_n} associated with the Hochschild arbor family t_n, namely the identity displayed in the paper for Σ_{n≥1}L_{t_n}(E,V)s^n.

Background

The paper defines a volume function for each arbor polytope Q_t by slicing according to the coordinate-sum height and encodes it through a Laplace transform L_t. For the Hochschild arbor family t_n, computer calculations suggest a closed generating series for these Laplace transforms. The proposed identity is explicitly described among the conjectural formulas and is left unresolved.