The Class of Purely Unrectifiable Sets in the Hilbert Space is coanalytic-complete

The space $F({\mathrm{\ell }}_2)$ of all closed subsets of ${\mathrm{\ell }}_2$ is a Polish space. We show that the subset $P\subset F({\mathrm{\ell }}_2)$ consisting of the purely 1-unrectifiable sets is $\text{Pii}$-complete.