Growth problem for the characteristic-zero Fibonacci Lie algebra

Let $K$ be a field of characteristic zero, let $R=K[t_i\mid i\ge 0]$, and let $L=\operatorname{Lie}(v_1,v_2)\subset\operatorname{Der}R$. Characteristic-zero growth problem. Determine the growth of the Lie algebra $L$. The paper notes that the growth is greater than polynomial and that $L$ is multigraded, but its precise asymptotic growth is left undetermined.