Consistency strength of indestructible supercompact maximality

Let $\text{ZFC}$ denote Zermelo–Fraenkel set theory with the axiom of choice. For a cardinal $\theta$, write $H(\theta)$ for the sets whose transitive closures have size less than $\theta$, and let $\text{MP}_{<ba\text{-dc}}(H(ba^+))$ denote the corresponding maximality principle. The theories considered below assert the existence of the indicated cardinals and ordinals.

Consistency-strength claim. The consistency strength of the theory $

$$\text{ZFC}+\text{“there is a supercompact }ba\text{, indestructible by }<ba\text{-directed closed forcing”}+\text{MP}_{<ba\text{-dc}}(H(ba^+))$$

is the same as that of the theory $

$$\text{ZFC}+\text{“}ba\text{ is supercompact”}+V_b4 \equiv V+ba<b4+\text{“}b4\text{ is inaccessible”}.$$

This identifies the consistency strength of the maximality principle together with indestructible supercompactness with a theory involving a supercompact cardinal and an inaccessible elementary-extension height. The supplied text gives no resolution status beyond the assertion itself.