We all know and love Cohen reals, and we can (and often do) define the Cohen forcing as partial functions $p\colon\omega\to 2$ with finite domain. The Prikry–Silver forcing is defined as partial functions $p\colon\omega\to 2$ with co-infinite domain.

These two couldn't be any more different. For example, Cohen reals are aggressively non-minimal, whereas Prikry–Silver reals are minimal.

We can look at a similar situation with other forcings that are given by finite conditions. For example $\operatorname{Col}(\omega,\omega_1)$ is a forcing notion whose conditions are finite partial functions $p\colon\omega\to\omega_1$. We can ask what would be the Prikry–Silver analogue of this forcing, then. That is, $\{p\colon\omega\to\omega_1\mid\operatorname{dom} p\text{ is co-infinite}\}$.

Interestingly, assuming CH this is the same as the standard collapsing forcing. This follows from the fact that the cardinality of the partial order is $2^{\aleph_0}$, which under CH is just $\aleph_1$, and we know that any forcing of size $\aleph_1$ which collapses $\omega_1$ is equivalent to $\operatorname{Col}(\omega,\omega_1)$.

Question.Is this so-called "Prikry–Silver collapse" provably equivalent to the standard collapsing forcing?