Higher-Hamiltonian eigenvalue conjecture for generalized Macdonald polynomials

Let $k\geq1$, let $\vec{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(N)})$ be an $N$-tuple of partitions, and let $H_k$ be the higher Hamiltonians with $H_1=X^{(1)}_0$ and

$$H_k=[X^{(1)}_{-1},[X^{(1)}_0,\ldots,[X^{(1)}_0,X^{(1)}_1]\ldots]],\qquad k\geq2,$$

where $X^{(1)}_0$ occurs $k-2$ times. Define

$$B^+_{\lambda}(z)=\frac{1-q^{\lambda_1-1}tz}{1-q^{\lambda_1}z}\prod_{i=1}^{\infty}\frac{(1-q^{\lambda_i}t^{-i}z)(1-q^{\lambda_{i+1}-1}t^{-i+1}z)}{(1-q^{\lambda_{i+1}}t^{-i}z)(1-q^{\lambda_i-1}t^{-i+1}z)}.$$

Higher-Hamiltonian eigenvalue conjecture.

$$e^{(k)}_{\vec{\lambda}}\overset{?}{=}\frac{(1-q)^{k-1}(1-t^{-1})^{k-1}}{1-p^{-1}}\oint\frac{dz}{2\pi\sqrt{-1}z}\prod_{i=1}^N B^+_{\lambda^{(i)}}(u_i z)z^{-k}.$$

The commuting Hamiltonians are expected to have these eigenvalues on the generalized Macdonald basis. The formula is presented as an expectation and is not established in the supplied text.