Non-solvable groups generated by involutions in which every involution is left $2$-Engel

The following problem is proposed as Problem 18.57 in [The Kourovka Notebook, No. 18, 2014] by D. V. Lytkina: Let $G$ be a finite $2$-group generated by involutions in which $[x, u, u] = 1$ for every $x \in G$ and every involution $u \in G$. Is the derived length of $G$ bounded? The question is asked of an upper bound on the solvability length of finite $2$-groups generated by involutions in which every involution (not only the generators) is also left $2$-Engel. We negatively answer the question.