Coxeter polynomial conjecture for non-negative posets of Dynkin type

Let $I$ be a poset with at most $13$ elements, and let $\mathrm{cox}_I(t)$ denote its Coxeter polynomial. Write $\mathrm{Dyn}_I$ for its Dynkin type, and call $I$ non-negative when its associated quadratic form is non-negative. A poset is connected when its Hasse diagram is connected.

Coxeter polynomial conjecture. If

$$\mathrm{cox}_I(t)=t^n+t^{n-1}+\cdots+t+1,$$

then $I$ is connected. If, additionally, $I$ is non-negative, then

$$\mathrm{Dyn}_I=\mathbb{A}_{|I|}.$$

The conjecture proposes that, for posets of at most $13$ elements, this cyclotomic Coxeter polynomial forces connectedness, and in the non-negative case identifies the Dynkin type as $\mathbb{A}_{|I|}$. It follows an example showing that the Coxeter polynomial alone does not preserve positivity of posets.