The Ordovician lattice structure conjecture

An Ordovician lattice is obtained from a Cambrian lattice by a finite sequence of mutations. For an Ordovician lattice $L$, let $a,b$ be atoms, let $\mu_i(L)$ denote mutation at an atom $i$, and let $\overrightarrow{B}$ be the quiver corresponding to $L$.

Ordovician lattice structure conjecture. The following hold:

1. Every Ordovician order is a lattice. In particular, every Cambrian lattice is mutable. Moreover, $u(x)$ is obtained from $x$ by a rotation sending each vertex to the next vertex clockwise.
2. If $a,b$ are atoms of $L$, then $[0,a\vee b]$ is a $(2,m)$-polygon for some $m\geq 2$. Moreover, $L$ is polygonal: $[x,y\vee z]$ is a polygon for any $x\prec y,z$, and $[x\wedge y,z]$ is a polygon for any $x,y\prec z$.
3. If $L$ corresponds to $\overrightarrow{B}$ and $i\in L$ is an atom, then $\mu_i(L)$ corresponds to $\mu_i(\overrightarrow{B})$.

The supplied text gives no resolution. The statement packages structural, polygonality, and mutation-compatibility properties of Ordovician lattices.