The Kawazumi–Vespa stable cohomology conjecture for Aut(F_n)

For $p,q\ge0$, let $H^{p,q}=H^{\otimes p}\otimes(H^*)^{\otimes q}$, and let $\mathcal{H}$ and $\mathcal{E}$ be the wheeled PROPs defined from stable twisted cohomology and the corresponding Ext-groups. Let $\mathcal{C}_{\mathcal{P}_0^{\circlearrowright}}$ be the wheeled PROP freely generated by the wheeled completion of the operadic suspension of the non-unital commutative operad. Kawazumi–Vespa's conjecture. For non-negative integers $i,p,q$, and sufficiently large $n$, there is an isomorphism of $\mathbb{Q}[\mathfrak{S}_p\times\mathfrak{S}_q]$-modules

$$H^i(\operatorname{Aut}(F_n),H^{p,q})=\begin{cases}\mathcal{C}_{\mathcal{P}_0^{\circlearrowright}}(p,q)&(i=p-q),\\0&(i\ne p-q).\end{cases}$$

Equivalently, the stable morphism of wheeled PROPs $\varphi:\mathcal{H}\to\mathcal{E}$ is an isomorphism. The source attributes this conjecture to Kawazumi and Vespa; it remains open.