Exceptional Specht-module decomposition conjecture for $\alpha_k$

For $k\in\mathbb{N}$, let $\alpha_k$ be the partition defined earlier in the source, let $\alpha_k^R$ be its row-labelled constituent, let $\alpha_k^C=(2k+3,2k+1,2k-1,\dots,9,7,5)$, and let $w(\alpha_k)$ denote its weight. Write ${\bf S}_{-1}^{\mathbb{C}}(\alpha_k)$ for the Specht module and ${\bf D}_{-1}^{\mathbb{C}}(\mu)$ for the corresponding simple module, with grading shift $\langle\cdot\rangle$. Exceptional decomposition conjecture.

$$\mathbf{S}_{-1}^{\mathbb{C}}(\alpha_k)=\mathbf{D}_{-1}^{\mathbb{C}}(\alpha_k^R)\left\langle w(\alpha_k)/2\right\rangle\oplus\mathbf{D}_{-1}^{\mathbb{C}}(\alpha_k^C)\left\langle w(\alpha_k)/2\right\rangle.$$

The paper proves decomposability and one direct summand in the surrounding discussion, while this full two-summand decomposition is presented only as an expectation; the supplied text gives no later resolution.