Bracket-width conjecture for formal current Lie algebras

Let $\mathfrak g$ be a finite-dimensional simple Lie algebra over an algebraically closed field $G_a$ of characteristic zero. The bracket width of a Lie algebra is the supremum of the smallest numbers of Lie brackets needed to express elements of its derived algebra as sums of brackets. Consider the formal current Lie algebra $\mathfrak g\otimes_{G_a}G_a[[t]]$.

Bracket-width conjecture. The bracket width of $\mathfrak g\otimes_{G_a}G_a[[t]]$ is equal to $2$ if $\mathfrak g$ is of type $\mathrm A_n$ or $\mathrm C_n$ $(n\ge 2)$, and to $1$ otherwise.

The paper has already established the general upper bound that every current Lie algebra has bracket width at most $2$. This conjecture gives the expected exact value for formal power-series current algebras; the supplied text does not indicate that the remaining lower-bound and classification claims have been resolved.