Divisibility and symmetry conjecture for links in Hopf algebras with the dual Chevalley property

Let $H$ be a non-cosemisimple Hopf algebra over $\Bbbk$ with the dual Chevalley property. Let $\mathcal{M}$ denote the set of simple subcoalgebras of $H$, and let $\mathcal{P}$ be the relevant set of paths or links between simple subcoalgebras, with $\mid{}^{1}\mathcal{P}\mid$, $\mid{}^{\mathcal{C}}\mathcal{P}\mid$, and $\mid\mathcal{P}{}^{\mathcal{C}}\mid$ denoting the corresponding cardinalities for $\mathcal{C}\in\mathcal{M}$. Divisibility and symmetry conjecture. For every $\mathcal{C}\in\mathcal{M}$,

$$\mid{}^{1}\mathcal{P}\mid \big| \mid{}^{\mathcal{C}}\mathcal{P}\mid.$$

Moreover,

$$\mid{}^{\mathcal{C}}\mathcal{P}\mid=\mid\mathcal{P}{}^{\mathcal{C}}\mid.$$

The conjecture is proposed to guide further research on Hopf algebras with the dual Chevalley property; the supplied source does not state a resolution, so the divisibility and equality assertions remain open.