Framing-operator identity for level-r generalized Macdonald functions (Conjecture BH)

Prove that, for every integer r ≥ 1, in the level-(r,0) horizontal Fock representation ρ^{(r,n_0)} of the extended quantum toroidal gl(1) algebra, the framing operator F^⊥ acts on the plethystic exponential by the identity ρ^{(r,n_0)}(F^⊥)·exp(Σ_{k>0}((-1)^k/(k(1−q_2^k)))·Σ_{i=1}^r v_i^k·p_k(α^{(i)})) = exp(−Σ_{k>0}(1/(k(1−q_2^k)))·Σ_{i=1}^r p_k(α^{(i)})·[(u'_i)^k + (1−q_3^k)·Σ_{j=i+1}^r (u'_j)^k]), where u'_i = (−)^{−1} u_i v_i, q_3=(q_1 q_2)^{−1}, and p_k(α^{(i)}) denotes the power-sum symmetric function in the i-th alphabet. Establish this identity for all degrees and all levels r.

Background

This conjecture (labelled Conjecture BH in the paper) gives a closed-form plethystic identity for the action of the framing operator F^⊥ on a specific exponential generating function built from the power-sum symmetric functions in r alphabets and the representation weights (u_i, v_i).

It is a central component in the authors’ extension of the Garsia–Haiman–Tesler (GHT) identity to higher levels, and its validity would yield key evaluation/normalization relations and underpin the complete algebraic proof of the higher-level GHT identity. The authors verified the identity computationally for low degrees and small r, and provided a complete proof at r=1, but the general case remains unproven.