Assume $\mathsf{ZF} + \mathsf{DC}$. Must there exist an injection from $\aleph_2$ to $\mathcal{P}(\mathbb{R})$? If not, what is the consistency strength of the nonexistence of such an injection?

I hope I'm not missing something obvious. Here are my thoughts so far:

- There is always a surjection from $\mathcal{P}(\mathbb{R})$ to $\aleph_2$.
- $\mathsf{ZFC}$, or more generally the existence of a wellordering of $\mathcal{P}(\mathbb{R})$, implies there is such an injection.
- $\mathsf{ZF} + \mathsf{AD}$ implies the existence of a surjection from $\mathbb{R}$ to $\aleph_2$, which implies there is such an injection.
- If there is an inner model of $\mathsf{ZFC}$ satisfying $(\aleph_1^V)^+ = \aleph_2^V$, for example in $V(\mathbb{R}^{V[G]})$ where $\kappa$ is inaccessible and $G \subset \operatorname{Col}(\omega,\mathord{<}\kappa)$ is a $V$-generic filter, then there is such an injection.