  We apply the Rasmussen spectral sequence to prove that the
$\mathbb{Z}^3$-graded vector space structure of the HOMFLYPT homology over
$\mathbb{Z}_2$ detects unlinks. Our proof relies on a theorem of Batson and
Seed stating that the $\mathbb{Z}^2$-graded vector space structure of the
Khovanov homology over $\mathbb{Z}_2$ detects unlinks.
