Stabilization conjecture for the filtered Kashiwara–Vergne Lie algebras

For each integer $k\geq 1$, let $\widehat{\mathfrak{krv}}^{(k)}_2$ and $\widehat{\mathfrak{krv}}^{(k+1)}_2$ denote the successive filtered Kashiwara–Vergne Lie algebras. Stabilization conjecture. For all $k\geq 1$,

$$\widehat{\mathfrak{krv}}^{(k)}_2=\widehat{\mathfrak{krv}}^{(k+1)}_2.$$

This is presented as an expected consequence of the conjectural decomposition of $\widehat{\mathfrak{krv}}_2$; the equality of all successive filtered terms remains open in the supplied text.