Lando's congruence for the universal $sl_2$ weight system

For each $ngeq 1$, let $\mathcal{D}_n$ be the $n$-chord diagram in which every chord intersects every other chord, and let $D_n=\varphi(\mathcal{D}_n)inmathbb{Z}[x]$. Write $(h_n)_{ngeq 0}=(1,1,2,7,38,295,\ldots)$ for the normalized median Genocchi numbers. Lando's conjecture. For all $ngeq 1$,

$$D_nequiv (-1)^{n-1}h_{n-1}xmod x^2.$$

The paper proves this particular case later as a consequence of its master theorem, so the original conjecture is solved.