If there is a bijection $\varphi:x\to y$ between two sets $x$ and $y$, we use the notation $x\simeq y$. The Weak Power Hypothesis is the following statement:

(WPH) For all sets $x, y$, whenever ${\cal P}(x) \simeq {\cal P}(y)$, then $x\simeq y$.

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

Does (WPH) imply the Boolean Prime Ideal Theorem (BPI)?