The genomic-tableaux upper-bound conjecture for Lagrangian Grassmannian constants

Let $Y={\rm OG}(n,2n+1)$ and $Z={\rm LG}(n,2n)$, with structure constants $b_{\lambda,\mu}^{\nu}$ and $c_{\lambda,\mu}^{\nu}$, respectively, indexed by strict partitions $\lambda$, $\mu$, and $\nu$. The genomic-tableaux upper-bound conjecture. For any strict partitions $\lambda$, $\mu$, and $\nu$,

$$|b_{\lambda,\mu}^{\nu}|\leq |c_{\lambda,\mu}^{\nu}|.$$

This extends the known cohomological relationship $c_{\lambda,\mu}^{\nu}=2^{\ell(\lambda)+\ell(\mu)-\ell(\nu)}b_{\lambda,\mu}^{\nu}$, where $\ell(\pi)$ is the number of nonzero parts of $\pi$. It holds in the cohomological case, has been verified computationally for $n\leq 6$, and is known when $\mu$ has a single part; the general conjecture remains open.