Generalised Franchetta conjecture for powers of hyperkähler varieties

Let $\mathcal{F}$ be the moduli stack of a locally complete family of polarised hyperkähler varieties, and let $\mathcal{X}\to\mathcal{F}$ be its universal family. For each $n\geq 1$, let $\mathcal{X}^{n/\mathcal{F}}$ denote the $n$-fold fiber product over $\mathcal{F}$. Generalised Franchetta conjecture. The family $\mathcal{X}^{n/\mathcal{F}}\to\mathcal{F}$ satisfies the Franchetta property for every $n\geq 1$. Thus generically defined cycles on every fiber power should inject into cohomology under the cycle class map. The source states both this conjecture and the ordinary Franchetta conjecture and describes them as open in general.