The genomic Q-tableaux upper-bound conjecture for symplectic Schubert constants

Let $Z={\rm LG}(n,2n)$ be the Lagrangian Grassmannian, and let $c_{\lambda,\mu}^{\nu}$ denote its Schubert structure constants for strict partitions $\lambda$, $\mu$, and $\nu$. Let $\mathtt{QBallot}_{\mu}(\nu/\lambda)$ be the set of ballot genomic $Q$-tableaux of shape $\nu/\lambda$ with content $\mu$. The genomic Q-tableaux upper-bound conjecture.

$$|c_{\lambda,\mu}^{\nu}|\leq \#\mathtt{QBallot}_{\mu}(\nu/\lambda).$$

The bound has been verified computationally for $n\leq 6$ and is sharp when $\mu$ has one part or when $|\nu|\leq |\lambda|+|\mu|$. The conjecture proposes a combinatorial upper bound for the absolute values of the symplectic Schubert structure constants.