This statement is known to be independent of ZFC: $ \def\qq{\mathbb{Q}} \def\pow{\mathcal{P}} \def\end{\mathrm{End}} \def\inj{\hookrightarrow} $
Given any sets $S,T$, if $\pow(S) ≈ \pow(T)$ then $S ≈ T$.
So I was wondering whether the following 'algebraic' statement is also independent of ZFC:
(★) Given any infinite-dimensional $\qq$-vector spaces $V,W$, if $\end(V)$ embeds into $\end(W)$ as a ring then $V$ embeds into $W$.
The motivation is to find some algebraic statement that has no set-theoretic flavour but is sensitive to set-theoretic assumptions.
I know that if GCH holds, then (★) also holds. Proof: For every $\qq$-vector space $V$ with basis $B$ we have that $\end(V)$ has the same cardinality as $(B×\qq)^B ≈ \qq^B$ (because $(B×\qq)^B$ $\inj (2×\qq)^{B×B}$ $≈ \qq^{B×B}$, and $B×B ≈ B$ if $B$ is infinite). Now take any infinite-dimensional $\qq$-vector spaces $V,W$ such that $\end(V)$ embeds into $\end(W)$. Let $B,C$ be bases for $V,W$ respectively. Then $\qq^B \inj \qq^C$ and so $2^B \inj \qq^B \inj \qq^C \inj 2^C$, and hence $B \inj C$ by GCH, yielding an embedding of $V$ into $W$.
But does ZFC already prove (★)? If not, is there some well-known set-theoretic axiom weaker than GCH that implies (★) over ZFC?
Recall that a family of elements $(e_i)_{i\in I}$ a ring $R$ is a family of orthogonal idempotents if $e_i^2=e_i$ for all $i\in I$ and $e_ie_j=0$ for all distinct $i,j\in I$. If $R=\operatorname{End}(V)$, then given such an family, each $e_i$ is a projection onto a subspace of $V$ and these subspaces are linearly independent (i.e., their sum is an internal direct sum). Conversely, given any decomposition of a subspace $V_0$ of $V$ as an internal direct sum and a projection $p$ from $V$ to $V_0$, the compositions of $p$ with the canonical projections of $V_0$ onto its direct summands form a family of orthogonal idempotents of $\operatorname{End}(V)$.
It follows that $\operatorname{End}(V)$ has a family of $\kappa$ nonzero orthogonal idempotents iff $\kappa\leq\dim V$. Such families are obviously preserved by injective ring homomorphisms, so if $\operatorname{End}(V)$ embeds in $\operatorname{End}(W)$, then $\dim V\leq \dim W$.