Interval conjecture for $(n-2,2)$ in the immersion poset

For a partition $\mu$ of $n$, let

$$A_\mu=\{\lambda\mid(1^n)\leqslant_I\lambda\leqslant_I\mu\}$$

be its lower interval in the immersion poset, and let $\lessdot_I$ denote a cover relation. The $(n-2,2)$ interval conjecture states that, for $n=5$ or $n\geqslant9$,

$$A_{(n-2,2)}=\{(1^n),(2,1^{n-2}),(2,2,1^{n-4}),(n-2,2)\},$$

with cover chain

$$(1^n)\lessdot_I(2,1^{n-2})\lessdot_I(2,2,1^{n-4})\lessdot_I(n-2,2).$$

Determining such intervals clarifies the immersion poset and the Schur-positivity of associated interval power sums; this claim is presented as a conjecture in the paper and remains open.