Proof-theoretic dilator conjecture for $0^\sharp$

Working over $\mathsf{ZFC}$ with the existence of $0^\sharp$, let $T$ be a $\Pi^1_3$-sound extension of $\mathsf{ACA}_0$. View $0^\sharp$ as a dilator.

Proof-theoretic dilator conjecture. The proof-theoretic ordinals satisfy

$$|T|_{\Pi^1_3}(0^\sharp)=|T+\exists 0^\sharp|_{\Pi^1_1[0^\sharp]}.$$

This proposes a proof-theoretic interpretation of $|T|_{\Pi^1_3}(P)$ for an $(n-1)$-ptyx with suitable definability, extending the paper’s main result beyond the $\Pi^1_2$ setting. The source gives no evidence of a resolution.