Let $G=SL(n)$ and $B$ be the Borel subgroup of $G$. $G/B$ is the complete flags variety $0\subset V_1 \subset \cdots \subset V_n=\mathbb{C}^n$. The Laumon space is the space $\mathrm{QMaps}^d(\mathbb{P}^1,G/B)$ of quasimaps from $\mathbb{P}^1$ to $G/B$, which can be identified with $0\subset \mathcal{V}_1 \subset \cdots \subset \mathcal{V}_n= \mathcal{O}^n_{\mathbb{P}^1}$ where $d_i=-\deg \mathcal{V}_i$.

As far as I understand, an Laumon space admits a description by a handsaw quiver given in section 2.3 of Finkelberg-Rybnikov. Negut showed that, up to suitable normalization, the generating function of equivariant cohomology of the Laumon spaces $\mathrm{QMaps}^d(\mathbb{P}^1,G/B)$ $(d=(d_1,\cdots,d_n)\ge0)$ is the eigenfunction of the quantum trigonometric Calogero-Sutherland hamiltonian.

My question is the following: for fixed $d=(d_1,\cdots,d_n)$, does the Givental's equivariant J-function (or I-function) of the Laumon space $\mathrm{QMaps}^d(\mathbb{P}^1,G/B)$ become an eigenfunction of an integrable differential equation?

Recently, physicists found a way to compute Givental's I-function by using supersymmetric gauge theory. (See this paper for example.) In principle, one can obtain the I-function of any GIT quotient space. Therefore, it would be great if one can see the integrable structure of the Laumon space in terms of I-function or J-function.