The recent computability theoretic analysis of the Cichoń diagram has introduced, for every cardinal characteristic A, a dual pair of classes of oracular strength called the highness class and non-lowness class for A. These classes can be relativized to any countable Turing ideal and, for any countable model V of ZFC, the inclusions and separations between these classes relativized to V correspond to the positions of the associated cardinals in the Cichoń diagram. In 2019 Greenberg, Kuyper and Turetsky asked what assumptions must be made on a countable Turing ideal I to obtain the "right" separations (all the expected inclusions hold for any Turing ideal). We show that for any countable model of Delta^1_1 comprehension I, three of these classes, namely the I-strongly null engulfing, I-strongly meagre engulfing, and I-dominating reals, are pairwise distinct. This improves on the previously known strength upper bound of ATR_0. To show this, we adapt ideas from bushy tree forcing to an infinitary context, introducing modifications of Laver and Hechler forcing which behave well with respect to hyperarithmetic definability. Joint work with Noam Greenberg.