Characterization conjecture for $\Sigma^1_2$-altitude

Let $R$ be a $\Sigma^1_2$-singleton real. Let $\mathsf{Alt}_{\Sigma^1_2}(R)$ denote its $\Sigma^1_2$-altitude, and consider transitive models $M$ of $\mathsf{ATR}_0^{\mathsf{set}}$ in which $R$ is a $\Sigma_1$-definable class in the language of set theory.

$\Sigma^1_2$-altitude characterization conjecture. $\mathsf{Alt}_{\Sigma^1_2}(R)$ is equal to the least height of such a transitive model $M$.

The $\Sigma^1_2$-altitude ranks $\Sigma^1_2$-singleton reals and arises as a byproduct of the main result. The source suggests this as a possible recursion-theoretic or set-theoretic characterization and gives no evidence of a resolution.