The intermediate-model classification conjecture for Magidor–Radin forcing

Let $G\subseteq \mathbb{M}[\vec{U}]$ be a $V$-generic filter, where

$$\forall\alpha\leq\kappa\; o^{\vec{U}}(\alpha)\leq\alpha.$$

Let $V\subseteq M\subseteq V[G]$ be a transitive $ZFC$ model. Intermediate-model classification conjecture. Either $M$ is a finite iteration of Magidor-like forcings as in the cited work, or there is a tree $T\subseteq[\kappa]^{<\omega}$ in $V$ with $ht(T)=\omega$ such that, for every $t\in T$ and every $\alpha\in\operatorname{Succ}_T(t)$, there is a name $\mathbb{M}[\vec{U}]^*_{t^{\smallfrown}\alpha}$ for a Magidor-like forcing. If $H$ is $V$-generic for the forcing that adds a branch through $T$ together with the forcings $\mathbb{M}[\vec{U}]^*_{t^{\smallfrown}\alpha}$ corresponding to that branch, then $M=V[H]$. The conjecture proposes a classification of all intermediate transitive $ZFC$ models between $V$ and the Magidor–Radin extension $V[G]$, distinguishing finite iterations from models generated by a tree-indexed sequence of Magidor-like forcings.