The combinatorial-model conjecture for shifted Littlewood–Richardson coefficients

Let $\lambda$ be a strict partition and let $\mu$ be a partition. The paper defines sets of pairs of tableaux, including $\overline{\mathcal{T}(\mu^t,\mu,s(\mathcal{T}_\lambda))}$ and $\overline{\mathcal{T}(\mu,\mu^t,s(\mathcal{T}_\lambda))}$, together with a map

$$\mathcal{S}^{\mu^t,\mu,\tilde{\lambda}}_{s(\mathcal{T}_\lambda),\mathcal{U}_\mu,\mathcal{U}_{\mu^t},s(\mathcal{T}_\lambda)}.$$

The combinatorial-model conjecture. The restriction of this map to $\overline{\mathcal{T}(\mu^t,\mu,s(\mathcal{T}_\lambda))}$ is a bijection onto $\overline{\mathcal{T}(\mu,\mu^t,s(\mathcal{T}_\lambda))}$. If $(U_\alpha^t,U_\beta)$ is not in $\mathcal{T}(\mu^t,\mu,s(\mathcal{T}_\lambda))$, then the corresponding image pair $(V_\alpha,V_\beta^t)$ is in $\mathcal{T}(\mu,\mu^t,s(\mathcal{T}_\lambda))$.

The stated combinatorial properties are intended to imply the squared inequality $g_{\lambda\mu}^2\leq c^{\tilde{\lambda}}_{\mu^t\mu}$. Their status is not resolved in the supplied text.