Let $G$ be a group and $x_1,\ldots,x_n,y_1,\ldots,y_n \in G$ involutions such that

$G = \langle x_1, \ldots , x_n \rangle = \langle y_1 , \ldots , y_n \rangle$

$g:=x_1 \cdots x_n = y_1 \cdots y_n$ is of finite order

Now assume that there exists $1 \leq k < n$ such that

$$ g = x_1 \cdots x_k y_{k+1} \cdots y_n.$$

Is it true that

$$ G = \langle x_1, \ldots , x_k , y_{k+1}, \ldots, y_n \rangle ?$$