The corank bound conjecture for non-negative posets of Dynkin type E

Let $I$ be a non-negative connected poset, let $|I|$ denote its cardinality, and let $\operatorname{\mathbf{crk}}_I$ denote its corank. Suppose that its Dynkin type is

$$\mathrm{Dyn}_I=\mathbb{E}_{|I|-\operatorname{\mathbf{crk}}_I}.$$

Corank bound conjecture. Then

$$\operatorname{\mathbf{crk}}_I\leq 3.$$

The paper establishes the corresponding classification for Dynkin type $\mathbb{A}$ and proves the bound $\operatorname{\mathbf{crk}}_I\leq 1$ there. Computations suggest that a uniform upper bound exists for the corank of posets of Dynkin type $\mathbb{E}_n$; this conjecture would imply $|I|\leq 11$ in the relevant setting.