Oblak's recursive conjecture for generic commuting nilpotent Jordan types

Let $P$ be a partition, let $\mathcal D_P$ be the associated poset, and let $C_a$ be a maximum-length U-chain in $\mathcal D_P$. Removing $C_a$ gives a partition $P'(P,a)$; recursively define $Ob(P)=(|C_a|,Ob(P'))$, with $Ob(P)=(n)$ when $P$ is almost rectangular. Oblak's recursive conjecture. The generic nilpotent Jordan-type map satisfies

$$\mathfrak Q(P)=Ob(P).$$

The source states that this conjecture appears close to resolved and cites subsequent work, but the supplied status is unknown; it is therefore recorded as open here.