Consider in $\newcommand{\ZF}{{\sf (ZF)}}\ZF$ the following statement:

Weak Power Hypothesis (WPH): if $X,Y$ are sets and there is a bijection between $\newcommand{\P}{{\cal P}}\P(X)$ and $\P(Y)$, then there is a bijection between $X$ and $Y$.

The Axiom of Dependent Choice states that

(DC): if $X$ is a set and $R\subseteq X\times X$ such that for all $a\in X$ there is $b\in X$ such that $(a,b)\in R$, then there is a function $s:\omega\to X$ such that $\big(s(n),s(n+1)\big) \in R$ for all $n\in \omega$.

Does (WPH) imply (DC)?

Note: It appears to be open whether (WPH) implies the much stronger Axiom of Choice (AC).