A decomposition multiplicity formula for $S_{\F_2}^{(n-4,2,1^2)}$

Let $n\geq 6$. For a partition $\lambda$ and a prime $p$, write $S_{\mathbb F_p}^{\lambda}$ for the corresponding Specht module and $D_{\mathbb F_p}^{\mu}$ for the simple module indexed by the $p$-regular partition $\mu$; $[S_{\mathbb F_p}^{\lambda}:D_{\mathbb F_p}^{\mu}]$ denotes its composition multiplicity. For a congruence condition, $\{n\con_p r\}$ is $1$ when $n\equiv r\pmod p$ and $0$ otherwise.

Multiplicity formula. The seven asserted formulas in the source hold: for $n\geq9$, $[S_{\mathbb F_2}^{(n-4,2,1^2)}:D_{\mathbb F_2}^{(n-4,4)}]=1$; for $n\geq7$, $[S_{\mathbb F_2}^{(n-4,2,1^2)}:D_{\mathbb F_2}^{(n-3,3)}]=\{n\con_2 0\}$; and the remaining five formulas are exactly those displayed in the source statement.

These are explicit decomposition-number assertions for symmetric-group Specht modules. The supplied text gives no resolution status or discussion establishing them, so their conjectural status requires verification.