$\DeclareMathOperator\supp{supp}$Let $f_i \in \bar{\mathbb{F}}_2[x_1,..,x_5]$ for $1 \leq i \leq 5$ be such that
$f_1(\bar{x}) = x_1 + x_5^2 + q_1$,
$f_2(\bar{x}) = x_2 + x_1^2 + q_2$,
$f_3(\bar{x}) = x_3 + q_3 $
$f_4(\bar{x}) = x_4 + q_4 $
$f_5(\bar{x}) = q_5 $
such that $$x_5^2 \notin \supp(q_1) \cup \supp(q_2) \cup \supp(q_3) \cup \supp(q_4) \cup \supp(q_5) $$ and $$ x_1^2 \notin \supp(q_2) \cup \supp(q_3) \cup \supp(q_4) \cup \supp(q_5).$$ If $x_i^{16} \in \bar{\mathbb{F}_2}[[f_1,...,f_5 ]]$ for each $ 1 \leq i \leq 5$, then can we say $$x_2^2 \in \supp(q_3) \cup \supp(q_4) \cup \supp(q_5)?$$ Note that $q_i$ are homogenous quadratics. This is a special case of trying to characterise the case $x_i^{2^r} \in \bar{\mathbb{F}}_2[[f_1,...,f_n]]$ for $1 \leq i \leq n$ for a quadratic system.
