Endpoint invariance conjecture for s-Cambrian congruences

Fix an $n$-tuple ${\boldsymbol{s}}$ of non-negative integers and an ${\boldsymbol{s}}$-arc $\alpha$. Let $\equiv_\alpha$ be the $\alpha$-Cambrian congruence of the ${\boldsymbol{s}}$-weak order, and let $W_{\boldsymbol{s}}/\equiv_\alpha$ be its $\alpha$-Cambrian lattice. Let the canonical join complex be the simplicial complex formed by canonical join representations of elements of this lattice, and let the ${\boldsymbol{s}}$-Cambrian foam and quotientoplex be the associated quotient polyhedral structures.

Endpoint invariance conjecture for s-Cambrian congruences. For fixed ${\boldsymbol{s}}$, the following depend only on the endpoints of $\alpha$: the cardinality of the $\alpha$-Cambrian lattice; the $f$-vector of its canonical join complex; the isomorphism class of its undirected cover graph; and the isomorphism class of the face lattice of the $\alpha$-Cambrian foam, equivalently of the $\alpha$-Cambrian quotientoplex.

This predicts that several combinatorial and polyhedral invariants are insensitive to the interior data of the arc. The source presents it as the main conjecture on ${\boldsymbol{s}}$-Cambrian congruences and gives no resolution.