Let $\mathbb P$ be a forcing that does not collapse $\omega_1$, $\theta$ sufficiently large and regular and $X\prec H_\theta$ a countable elementary substructure with $\mathbb P\in X$ as well as $p\in \mathbb P\cap X$. This is a typical situation if one deals with (semi)proper forcings. Let $\delta=X\cap\omega_1$. Assume further that $\dot S\in X$ is a $\mathbb P$-name for a subset of $\omega_1$. I would like to find a $(X, \mathbb P)$-semigeneric condition $q\leq p$ and simultaneously control whether $\delta\in\dot S^G$ or not. Lets say we ask for such $q$ so that $$q\Vdash \check\delta\notin\dot S$$ There are obvious obstructions to this: If there is $T\in \mathcal P(\omega_1)\cap X$ with $\delta\in T$ and $$p\Vdash\check T\subseteq \dot S \mod \mathrm{NS}_{\omega_1}$$ then there cannot be such $q$. I can prove that if the Strong Reflection Principle ($\mathrm{SRP}$) holds, this is the "only reason" for the nonexistence of such $q$:

Assume $\mathrm{SRP}$ holds. Let $\mathbb P, X, p, \dot S$ as above. Exactly one of the following holds:

- There is $q\leq p$ that is $(\mathbb P, X)$-semigeneric and $q\Vdash\check\delta\notin \dot S$
- There is $T\in \mathcal P(\omega_1)\cap X$ with $\delta\in T$ and $$p\Vdash\check T\subseteq \dot S \mod \mathrm{NS}_{\omega_1}$$

The proof of $\neg 1.\Rightarrow 2.$ is similar to the the proof of "stationary set preserving forcings are semiproper" (under $\mathrm{SRP}$). Let us say the name dichotomy holds for $\mathbb P$ if exactly one of 1. and 2. holds for any $X, p, \dot S$ as above. One can hope that if a property is provable for the class of $\omega_1$-preserving forcings under $\mathrm{SRP}$ then maybe this property holds in general for more "well-behaved" classes of forcings. So my question is:

Is it provable (in $\mathrm{ZFC}$) that the name dichotomy holds for

- $\sigma$-closed forcings?
- proper forcings?
- semiproper forcings?