I discover an injection proof nearly at the same time: 
- If $w$ is odd ($w=2\ell-1$):
        \begin{equation*}
        \begin{split}
        \hspace{-8.4ex}\underbrace{(\blacksquare\Box\Box)}_{\mbox{splitted into each other boxes}}\hspace{-8.4ex}(\Box\blacksquare\Box)(\Box\Box\Box)(\Box\Box\blacksquare)\mapsto(\blacksquare\Box\blacksquare\Box)(\Box\Box\Box\Box)(\Box\Box\Box\blacksquare)
        \end{split}
        \end{equation*}
- If $w$ is even ($w=2\ell$):
        \begin{equation*}
        \begin{matrix}
        \hspace{-7.5ex}\underbrace{(\blacksquare\Box\blacksquare\Box)}_{\mbox{splitted into each other boxes}}\hspace{-7.5ex}(\Box\blacksquare\Box\Box)(\blacksquare\Box\Box\blacksquare)(\blacksquare\Box\blacksquare\Box)(\blacksquare\blacksquare\Box\Box)\\
        \downarrow\\
        (\blacksquare\Box\blacksquare\Box\Box)(\Box\blacksquare\Box\Box\blacksquare)\hspace{-1.9ex}\underbrace{(\blacksquare\blacksquare\Box\blacksquare\Box)}_{\mbox{flip each position}}\hspace{-1.5ex}(\Box\blacksquare\blacksquare\Box\Box)\\
        \downarrow\\
        (\blacksquare\Box\blacksquare\Box\Box)(\Box\blacksquare\Box\Box\blacksquare)(\Box\Box\blacksquare\Box\blacksquare)(\Box\blacksquare\blacksquare\Box\Box)\\
        \end{matrix}  
        \end{equation*}