  We prove that, for $1 \le p<q<\infty$, the partially ordered set
$P(\omega)/{\rm Fin}$ can be embedded into Borel equivalence relations between
$\mathbb{R}^\omega/\ell_p$ and $\mathbb{R}^\omega/\ell_q$. Since there is an
antichain of size continuum in $P(\omega)/{\rm Fin}$, therefore there are
continuum many incomparable Borel equivalence relations between
$\mathbb{R}^\omega/\ell_p$ and $\mathbb{R}^\omega/\ell_q$.
