The Shuffle conjecture

Let $n$ be a positive integer, let $\mu=(\mu_1,\mu_2,\ldots,\mu_k)\vdash n$, and let $E_1,E_2,\ldots,E_k$ be successive segments of the word $123\cdots n$ of respective lengths $\mu_1,\mu_2,\ldots,\mu_k$. Let ${\cal PF}_n$ denote the collection of Parking Functions in the $n\times n$ square, let $\sigma(PF)$ be the reading word of $PF$, and let $\shuffle$ denote shuffling. The Shuffle conjecture.

$$\big\langle \nabla e_n,h_\mu\big\rangle=\sum_{PF\in{\cal PF}_n}t^{\operatorname{area}(PF)}q^{\operatorname{dinv}(PF)}\chi\big(\sigma(PF)\in E_1\shuffle E_2\shuffle\cdots\shuffle E_k\big).$$

This conjecture gives a combinatorial interpretation of the scalar products of $\nabla e_n$ with homogeneous symmetric functions and, in particular, predicts the $(t,q)$-enumeration of Parking Functions with prescribed shuffle reading words. Its specialization at $t=1/q$ implies the additional $q$-enumeration stated below, which the paper describes as open.