A variety of partially conservative sentences

We study the existence of a $\mathrm{\Theta }$ sentence which is simultaneously $\mathrm{\Gamma }$-conservative over consistent RE extensions $T$ and $U$ of Peano Arithmetic for various reasonable pairs $(\mathrm{\Gamma },\mathrm{\Theta })$. As a result of this study, we prove the existence of a sentence which is essentially $\mathrm{\Theta }$ and exactly hereditarily $\mathrm{\Gamma }$-conservative over any single theory $T$ for various reasonable pairs $(\mathrm{\Gamma },\mathrm{\Theta })$. This is an affirmative answer to Guaspari’s question.