Index equality for periodic contractions of semisimple Lie algebras

Let $\mathfrak g$ be a semisimple Lie algebra and let $\vartheta$ be a periodic automorphism of order $m$, inducing the grading

$$\mathfrak g=\bigoplus_{i=0}^{m-1}\mathfrak g_i.$$

Let $\mathfrak g_{(0)}$ be the contraction of $\mathfrak g$ whose bracket agrees with the original bracket on pairs $\mathfrak g_i,\mathfrak g_j$ with $i+j\leq m-1$ and is zero when $i+j>m-1$. Here $\operatorname{ind}\mathfrak a$ denotes the index of a Lie algebra $\mathfrak a$. Index-equality conjecture. For any periodic automorphism $\vartheta$, one has

$$\operatorname{ind}\mathfrak g_{(0)}=\operatorname{ind}\mathfrak g.$$

By semicontinuity, the left-hand side is always at least the index of $\mathfrak g$, and equality is known in several cases, including involutions and when $\mathfrak g_1$ contains a regular element of $\mathfrak g$. The general case is left open by the source.