I'm wondering if a particular theory of second order arithmetic has been studied or is known to be equivalent to some other theory.

Consider the formulas generated by $\Pi^1_1$ and $\Sigma^1_1$ formulas by propositional combinations (the title refers to this, rather informally, as the difference hierarchy, since these formulas are essentially the difference between two $\Pi^1_1$ formulas, the difference between two such, and so on). Let's call this class of formulas $\Delta$, for convenience.

My question is what the strength (either reverse mathematical or proof theoretic) of $\Delta-TI_0$, the theory which allows transfinite induction along any well-ordering for $\Delta$ formulas, is. This has proof theoretic strength strictly greater than $\Pi^1_2-TI_0$ (Rathjen and Weierman's proof that $\Pi^1_2-TI_0$ implies well-foundedness of $\psi\Omega^{\Omega^n}$ goes through, and furthermore this theory can carry out the key induction internally), and strictly less than $\Pi^1_1-CA_0$ (the standard argument that $\Pi^1_1-CA_0$ proves the existence of a $\beta$-model of $\Pi^1_\infty-TI_0$ suffices).