I discover an injection proof, intuitively:

• 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*}

I discover an injection proof.

Intuitively:

• 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*}

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*}

I discover an injection proof, intuitively:

• 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*}