The d-complete poset hook-length bijection conjecture

Let $P$) be a $d$-complete poset and let $c$ be a minimal element. Define $A_c(G)=\{z\in P:c\in\overline H_z\}$ for $G\in\mathcal{G}_c$, and let $\mathcal{G}_c^k$ denote the elements of $\mathcal{G}_c$ having exactly $k$ fixed points. There exist maps

$$s=s_c:\mathcal{P}(A_c)\to P,$$

extended to $s:\mathcal{G}_c\to P$ by $s(G)=s(A_c(G))$, and a map $e=e_c:\mathcal{G}_c\to\mathcal{G}_c$ satisfying the following properties.

The d-complete poset hook-length bijection conjecture. If $G\in\mathcal{G}_c^0$, then $s(G)=c$ and $e(G)=G$; if $G\in\mathcal{G}_c^k$ for $k\geq1$, then $e(G)\in\mathcal{G}_c^{k-1}$. If $G\in\mathcal{G}_c^k$ for $k\geq1$, then $e(G)_{s(G)}=s(e(G))$; moreover, if $z\not\geq s(G)$, then $e(G)_z=G_z$. Finally, given $G'\in\mathcal{G}_c^k$ and $z$ such that $G'_z=s(G')$, there is exactly one $G\in\mathcal{G}_c^{k+1}$ satisfying $s(G)=z$ and $e(G)=G'$.

These maps would provide a bijective proof of the hook-length formula for $d$-complete posets by organizing the relevant objects according to their number of fixed points. The conjecture concerns the existence of the stated raising and lowering structure and remains open in the source.