On extensions of partial $n$-quasigroups of order 4

We prove that every collection of pairwise compatible (nowhere coinciding) $n$-ary quasigroups of order 4 can be extended to an $(n+1)$-ary quasigroup. In other words, every Latin $4\times\cdots\times4\times l$-parallelepiped, where $l=1,2,3$, can be extended to a Latin hypercube.