Nonexistence conjecture for infinite fields with indecomposable multiplicative groups

Let $F$ be a field, and write $F^\times$ for its multiplicative group. A group is indecomposable if it cannot be expressed as a nontrivial direct product of groups. Nonexistence conjecture. There is no infinite field $F$ whose multiplicative group $F^\times$ is indecomposable.

The finite fields with indecomposable multiplicative groups are known; the conjecture concerns whether any infinite field can have this property. A negative answer would solve the longstanding realizability problem for indecomposable abelian groups in the setting of field multiplicative groups.