Essential dimension of sequences of quaternion algebras

We study the essential dimension of the set of isometry classes of $m$-tuples $(\varphi_1,...,\varphi_m)$ of quadratic $n$-fold Pfister forms over a field $F$ such that the Witt class of $\varphi_1 \perp \ldots \perp \varphi_m$ 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 $n=\text{char} F=2$, $m=4$.