Semisimplicity conjecture for the Specht modules indexed by $\beta_k$

For odd $k\in2\mathbb{N}+1$, let $\beta_k$ be the partition defined earlier in the source, and let $\beta_k^R$ and $\beta_k^C$ be the two labels specified there. Let $w(\beta_k)$ denote the weight, ${\bf S}_{-1}^{\mathbb{C}}(\beta_k)$ the Specht module, and ${\bf D}_{-1}^{\mathbb{C}}(\mu)$ the corresponding simple module, with grading shift $\langle\cdot\rangle$. Semisimplicity conjecture for $\beta_k$.

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

The surrounding theorem establishes decomposability and a direct summand, while the displayed equality is explicitly only predicted; the supplied text gives no resolution status.