  Based on results by Dani\v{l}\v{c}enko, in 1987 Burris and Willard have
conjectured that on any $k$-element domain where $k\geq 3$ it is possible to
bicentrically generate every centraliser clone from its $k$-ary part. Later,
for every $k\geq 3$, Snow constructed algebras with a $k$-element carrier set
where the minimum arity of the clone of term operations from which the
bicentraliser can be generated is at least $(k-1)^2$, which is larger than $k$
for $k\geq 3$.
  We prove that Snow's examples do not violate the Burris-Willard conjecture
nor invalidate the results by Dani\v{l}\v{c}enko on which the latter is based.
We also complement our results with some computational evidence for $k=3$,
obtained by an algorithm to compute a primitive positive definition for a
relation in a finitely generated relational clone over a finite set.
