Weak and strong good-model conjectures for semistable surface families

Let $f:S\to\Delta$ be a semistable family of surfaces, endowed with a line bundle $\mathcal L$, and let $\delta$ be a positive integer. A semistable model is $\delta$-absolutely good when its relative Severi variety equals its regular part as cycles; the family is $\delta$-well behaved when it has a $\delta$-good model. Good-model conjecture. Under suitable assumptions, $f:S\to\Delta$ is $\delta$-well behaved (weak version), and under suitable assumptions it has a $\delta$-absolutely good semistable model (strong version). The paper proves well-behavedness for particular degenerations, but the assumptions guaranteeing either assertion are left to be discovered.