Six-cycle spanning conjecture for the F4 diagrammatic category

Let $\mathpzc{F}$ be the diagrammatic category for type $F_4$, with generating object ${\mathsf{I}}$. A graph is component-planar if each of its connected components is a planar graph. For $m,n\in\mathbb{N}$, consider component-planar graphs from ${\mathsf{I}}^{\otimes m}$ to ${\mathsf{I}}^{\otimes n}$ whose cycles all have length at least six.

Six-cycle spanning conjecture. For $m,n\in\mathbb{N}$, these component-planar graphs span

$$\mathpzc{F}({\mathsf{I}}^{\otimes m},{\mathsf{I}}^{\otimes n}).$$

This proposes a restricted spanning family for every hom-space in the diagrammatic category, replacing graphs with shorter cycles by linear combinations of graphs satisfying the six-cycle condition. The source presents this as a conjecture; no resolution is supplied.