The type $C^{(1)}_\ell$ Garnir-element conjecture for Specht modules

Let $\lambda\in \mathscr{P}^l_n$, and let $A\in [\lambda]$ be a Garnir node. Suppose $\mathbf{B}^A$ contains $a$ bricks in the first row and $b$ bricks in the second row. For each permutation $u$ of the bricks, let $\tau^A_u$ denote the corresponding product of the brick-transposition elements, and let $w^{\mathtt{T}^A}$ be the permutation taking the dominant tableau $\mathtt{T}^A$ to the relevant tableau. Let $g^\lambda_A$ be the Garnir element, $m^\lambda$ the cyclic generator, and $\mathcal{S}^\lambda$ the Specht module. The type $C^{(1)}_\ell$ Garnir-element conjecture. In type $C^{(1)}_\ell$,

$$g^\lambda_A=\sum_u \tau^A_u\psi_{w^{\mathtt{T}^A}}m^\lambda,$$

where the sum is over all $u\in\mathfrak{S}_{a+b}/(\mathfrak{S}_a\times\mathfrak{S}_b)$. Moreover, the stated theorem and corollary on Specht modules hold in type $C^{(1)}_\ell$, yielding a homogeneous basis of $\mathcal{S}^\lambda$ indexed by standard $\lambda$-tableaux. This extends the corresponding Garnir-element formula and homogeneous-basis result to affine type $C$; the source provides no evidence that the assertion has been proved or disproved.