Geometric bounded-components form of the arithmetic Shafarevich conjecture

Let $C'$ be a semistable curve over $F$, and let $C_L$ be the model associated with a finite nonramified extension $L$ of $F(C')$. Geometric Shafarevich conjecture. There is a number $J(F(C'))$ such that the number of components of every fiber of $C_L$ is bounded by $J(F(C'))$ for every finite extension $L$ of $F(C')$ contained in $F(C')^{\mathrm{nr}}$. This is presented as a geometric reformulation of the preceding arithmetic conjecture; the source gives no resolution status.