2-affine complete algebras need not be affine complete

For each $k \in \N$, we exhibit a finite algebra $\ab{R}_k$ such that $\ab{R}_k$ is $k$-affine complete, but not $(k+1)$-affine complete; this means that every $k$-ary congruence preserving function on $\ab{R}_k$ lies in $\Pol_k \ab{R}_k$, but there is a $(k+1)$-ary congruence preserving function of $\ab{R}_k$ that does not lie in $\Pol_{k+1} \ab{R}_k$.