By famous KPT theorem, if $\mathsf{T^i_2}\vdash \mathsf{T^j_2}$ for some $0<i<j$, then the polynomial hierarchy collapses. So it seems this bounded arithmetic hierarchy does not collapse. Also, conservative results for some classes of formulas lead to collapsing complexity classes which is not expected. My question is about the $ \mathsf{T^i_2+EXP}$ theories.

Q1. What is the relationship between $\mathsf{T^i_2+EXP}$ and $\mathsf{T^j_2+EXP}$ for $i<j$? Is it true that $\mathsf{T^i_2+EXP\vdash T_2}$?

Separating these theories implies separating bounded arithmetics which it seems to be hard.

Q2. What about conservative results? For example does $\mathsf{T^i_2+EXP}$ proves every $\Pi_2$ sentences of $\mathsf{T_2+EXP}$?

Thanks in advance.