Existence of Frobenius structures for non-general multiplets

Let $A=(a_1,ots,a_r)$ be a non-general multiplet. A Frobenius structure is understood to have the conditions (i), (ii), (iii), (v), and (vi) of the paper's Theorem 1; condition (iv) is the additional condition referred to below.

Existence conjecture. For each non-general multiplet $A$, there exists a Frobenius structure satisfying conditions (i), (ii), (iii), (v), and (vi) of Theorem 1, but not condition (iv).

This conjecture concerns the expected distinction between non-general multiplets and the semi-general cases treated in the preceding argument. The supplied text gives no resolution status.