Write $\mathrm{ICF}$ for the "injective continuum function hypothesis" i.e. the sentence of ZFC expressing that

$$2^X \cong 2^Y \rightarrow X \cong Y$$

for all sets $X$ and $Y$, where $\cong$ denotes equipotency.

If I understand correctly:

1. $\mathrm{ICF}$ is independent of $\mathrm{ZFC}$.
2. $\mathrm{GCH}$ implies $\mathrm{ICF}.$ (In the presence of the $\mathrm{ZFC}$ axioms.)
3. The converse of 2 is false.

**Question.** Are there situations outside set theory (e.g. group theory, measure theory, etc.) where it would be useful if $\mathrm{ICF}$ were true, irrespective of whether or not $\mathrm{GCH}$ is true?