Minuscule P-partition rowmotion orbit conjecture

Let

$$(P,Q) \in \{(\Lambda_{\mathrm{Gr}(k,n)},T_{k,n}),(\Lambda_{\mathrm{OG}(6,12)},\Phi^+(H_3)),(\Lambda_{\mathbb{Q}^{2n}},\Phi^+(I_2(2n)))\}$$

be a minuscule doppelgänger pair, and let $\mathrm{PP}^{\ell}(P)$ denote the $P$-partitions of height $\ell$.

Minuscule P-partition rowmotion orbit conjecture. For every $\ell\geq 1$, there is a bijection $\varphi$ between the $\mathrm{row}$-orbits of $\mathrm{PP}^{\ell}(P)$ and those of $\mathrm{PP}^{\ell}(Q)$ such that, for each orbit $\mathcal O\subseteq\mathrm{PP}^{\ell}(P)$,

$$\#\mathcal O=\#\varphi(\mathcal O),$$

and

$$\sum_{T\in\mathcal O}\mathrm{ddeg}(T)=\sum_{T\in\varphi(\mathcal O)}\mathrm{ddeg}(T).$$

The conjecture was checked computationally for several small cases, including $\ell=2,3,4$, but the paper states that the known doppelgänger bijection does not commute with rowmotion when $\ell>1$ and offers no proof.