VEST is W[2]-hard

In this short note, we show that the problem of VEST is $W[2]$-hard for parameter $k$. This strengthens a result of Matoušek, 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 $k$.