Hodges's conjecture on relative categoricity and the Gaifman property

Let $T$ be a complete theory with quantifier elimination in a relational language $L$, and let $P$ be a distinguished unary predicate symbol in $L$. For a model $M\models T$, write $M^{P}$ for the $L$-substructure of $M$ with universe $P(M)$, and let $T^{P}$ be the common complete theory of these structures as $M$ ranges over models of $T$. The theory $T$ is relatively categorical if every isomorphism between the $P$-parts of two models of $T$ lifts to an isomorphism between the models. Hodges's conjecture. If $T$ is relatively categorical, then $T$ has the Gaifman property. The source indicates that this conjecture is refuted by a counterexample, so the claim is not valid in general.