Broadhurst–Kreimer conjecture for the linearized Kashiwara–Vergne Lie algebra

Let $\widehat{\mathfrak{lkv}}_2$ be the linearized Kashiwara–Vergne Lie algebra, bigraded by weight and depth. Let $\mathrm{BK}_{W,D}$ be the coefficients defined by

$$\sum_{W,D} s^W t^D \mathrm{BK}_{W,D}:=\operatorname{Log}\left(\frac{1}{1-\frac{s^3t}{1-s^2}-\frac{s^{12}(t^2-t^4)}{(1-s^4)(1-s^6)}}\right),$$

with $s,t$ formal variables and

$$\operatorname{Log}(f)(s,t)=\sum_{\ell\geq 1}\frac{\mu(\ell)}{\ell}\log(f(s^\ell,t^\ell)),$$

where $\mu$ is the Möbius function. Broadhurst–Kreimer conjecture for $\widehat{\mathfrak{lkv}}_2$. The dimension of the bigraded component $\widehat{\mathfrak{lkv}}_2^{(W,D)}\subset\widehat{\mathfrak{lkv}}_2$ of weight $W$ and depth $D$ is $\mathrm{BK}_{W,D}$. This is the natural Kashiwara–Vergne analogue of the Broadhurst–Kreimer prediction for $\mathfrak{lds}$; the supplied text gives no resolution status.