The genomic Q-tableaux dagger lower-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. Let $\mathtt{QBallot}_{\mu}(\nu/\lambda)$ be the set of ballot genomic $Q$-tableaux of shape $\nu/\lambda$ with content $\mu$, and define $\mathtt{QBallot}^{\dagger}_{\mu}(\nu/\lambda)$ to be the subset consisting of tableaux for which no gene contains both primed and unprimed labels. The genomic Q-tableaux dagger lower-bound conjecture.

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

Together with the preceding upper bound, this conjecture would give combinatorially related lower and upper bounds for the absolute values of the structure constants. The statement has been computer verified for $n\leq 6$ in the surrounding discussion, but the general assertion remains open.