Space-efficient RLZ-to-LZ77 conversion

Consider a text $T[\mathrm{1..}n]$ prefixed by a reference sequence $R=T[\mathrm{1..}\mathrm{\ell }]$. We show how, given $R$ and the $z^{\mathrm{\prime }}$-phrase relative Lempel-Ziv parse of $T[\mathrm{\ell }+\mathrm{1..}n]$ with respect to $R$, we can build the LZ77 parse of $T$ in $n\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{l}\mathrm{o}\mathrm{g}(n)$ time and $O(\mathrm{\ell }+z^{\mathrm{\prime }})$ total space.