Modular transform of free fermion generalised Gibbs ensembles and generalised power partitions

In [1] a conjecture for the modular transformation of the free fermion generalised Gibbs ensemble (GGE) was given where only the KdV charge associated to the weight four quasi primary field was inserted. In this paper we first generalise this conjecture to the case with an arbitrary, finite collection of KdV charges in the GGE. These GGEs are generalisations of the generating function of power partitions. We prove the conjectured transformation for the case with a finite number of charges inserted and discuss the case with an infinite number of charges.