Essential dimension of sequences of quadratic Pfister forms

We study the essential dimension of the set of isometry classes of $m$-tuples $({\phi }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\phi }_m)$ of quadratic $n$-fold Pfister forms over a field $F$ such that the Witt class of ${\phi }_1\perp \dots \perp {\phi }_m$ lies in $I_q^{n+1}F$. We show that the essential dimension is equal to $n+1$, when $m=3$, and is either $4$ or $5$, when $n=\text{char}F=2$, $m=4$.