Equivalence of short pp-definitions and few subpowers

Let $\Gamma$ be a constraint language. It has short pp-definitions if every $n$-ary relation pp-definable from $\Gamma$ is definable by a primitive positive formula whose length is bounded by a polynomial in $n$. It has few subpowers if the number of $n$-ary relations pp-definable from $\Gamma$ is bounded by $2^{p(n)}$ for some polynomial $p$. Short pp-definitions conjecture. A constraint language has short pp-definitions if and only if it has few subpowers. A cardinality argument establishes the implication from short pp-definitions to few subpowers; the converse is conjectured and is verified in the paper for a large subclass, including all constraint languages on three-element domains.