The mirror image of the language of 2-synchronizing words

A word $w$ over a finite alphabet $\Sigma$ is called $n$-synchronizing if for each deterministic finite automaton $\mathscr A=\langle Q,\Sigma,\delta\rangle$ such that $|Q|=n+1$ the equality $|\delta(Q,w)|=1$ holds provided that $|\delta(Q,u)|=1$ for some word $u\in\Sigma^*$ (depending on $\mathscr A$). In this paper we prove that the language of all 2-synchronizing words is closed under the mapping that associates each word $w=a_1a_2\cdots a_t\in\Sigma^*$ with its mirror image $\overleftarrow w=a_t\cdots a_2a_1$.