Iterates of $M_1$

Assume $\boldsymbol{\Delta}^1_{2}$-determinacy. Let $L_{\kappa_3}[T_2]$ be the admissible closure of the Martin-Solovay tree and let $M_{1,\infty}$ be the direct limit of $M_1$ via countable trees. We show that $L_{\kappa_3}[T_2] \cap V_{u_{\omega}} = M_{1,\infty} | u_{\omega}$.