  In this short note, we show that the problem of VEST is $W[2]$-hard for
parameter $k$. This strengthens a result of Matou\v{s}ek, who showed
$W[1]$-hardness of that problem. The consequence of this result is that
computing the $k$-th homotopy group of a $d$-dimensional space for $d > 3$ is
$W[2]$-hard for parameter $k$.
