Decomposability conjecture for multiplicative groups of infinite fields

Let $F$ be an infinite field, and write $F^{\times}$ for its multiplicative group. A group is decomposable if it is isomorphic to a direct sum of two nontrivial subgroups. Decomposability conjecture. Every infinite field has a decomposable multiplicative group:

$$F^{\times}\cong A\oplus B$$

for some nontrivial groups $A$ and $B$. The theorem preceding this conjecture proves the assertion for infinite fields whose characteristic is not $2$; the conjecture concerns the remaining characteristic-$2$ case.