Horn rewritability vs PTime query evaluation for description logic TBoxes

We study the following question: if τ is a TBox that is formulated in an expressive DL L and all CQs can be evaluated in PTime w.r.t. τ, can τ be replaced by a TBox τ' that is formulated in the Horn-fragment of L and such that for all CQs and ABoxes, the answers w.r.t. τ and τ' coincide? Our main results are that this is indeed the case when L is the set of ALCHI or ALCIF TBoxes of quantifier depth 1 (which covers the majority of such TBoxes), but not for ALCHIF and ALCQ TBoxes of depth 1