Converse root subgroup factorization conjecture for loop group elements

Let $g_1$ and $g_2$ be elements associated with the sets of loop group elements satisfying conditions (I.1), (I.3), (II.1), and (II.3), respectively, in the theorem above. A root subgroup factorization is a factorization of the form described in (I.2) for $g_1$, or in (II.2) for $g_2$.

Converse root subgroup factorization conjecture. If $g_1$ is in the identity connected component of the sets in (I.1) and (I.3), then the converse holds: $g_1$ has a root subgroup factorization as in (I.2). If $g_2$ is in the identity connected component of the sets in (II.1) and (II.3), then the converse holds: $g_2$ has a root subgroup factorization as in (II.2).

This proposes converses to the implications established in the preceding theorem, under an identity-connected-component hypothesis. The supplied text gives no resolution of these assertions.