Grothendieck's perfectness conjecture for the Néron component-group pairing

Let $K$ be a complete discrete valuation field with ring of integers $\mathcal{O}_{K}$ and perfect residue field $k$ of characteristic $p>0$. Let $A$ be an abelian variety over $K$, let $\mathcal{A}$ be its Néron model over $\mathcal{O}_{K}$, and let $\mathcal{A}_{x}$ be the special fiber over $x=\operatorname{Spec} k$. Write $A^{\vee}$ for the dual abelian variety, with corresponding Néron model $\mathcal{A}^{\vee}$ and special fiber $\mathcal{A}_{x}^{\vee}$. The component groups $\pi_{0}(\mathcal{A}_{x})$ and $\pi_{0}(\mathcal{A}_{x}^{\vee})$ are finite étale group schemes over $k$, and Grothendieck's canonical pairing is

$$\pi_{0}(\mathcal{A}_{x}^{\vee})\times\pi_{0}(\mathcal{A}_{x})\to\mathbb{Q}/\mathbb{Z}.$$

Grothendieck's perfectness conjecture. The pairing above is perfect.

This pairing is the obstruction to extending the Poincaré biextension from $A^{\vee}\times A$ to the Néron models. The conjecture is proved in the paper by reducing from the known semistable case via Galois descent, so its status is solved.