The Kinna–Wagner conjecture for intermediate generic extensions

Let $V$ be a model satisfying the Kinna–Wagner principle $\mathsf{KWP}$, let $G$ be a $V$-generic filter, and let $M$ be an intermediate model with $V\subseteq M\subseteq V[G]$. Kinna–Wagner conjecture. If $M$ satisfies $\mathsf{KWP}$, then there is a set $x$ such that

$$M=V(x).$$

This conjecture proposes a structural description of intermediate models that retain the Kinna–Wagner principle. The source gives no resolution; it follows a discussion of uniform failure or satisfaction of $\mathsf{KWP}$ across generic and symmetric multiverses.