The hyperkähler period-index conjecture is false

Huybrechts conjectured that for every Brauer class $α$ on a hyperkähler variety $X$ we have that $\mathop{\rm ind}(α)\mid\mathop{\rm per}(α)^{\dim(X)/2}$, strengthening the usual period-index conjecture. We show that this fails on certain hyperkähler fourfolds, both in type $\mathrm{K}3^{[2]}$ and $\mathrm{Kum}^2$.