Verification of the functional-unit independence assumption for the five-parameter quintic family

Verify part (1) of Assumption 1.1 for the five-parameter family (x-A(t)y)(x-B(t)y)(x-C(t)y)(x-D(t)y)(x-E(t)y)+y^5=q, and thereby establish whether the family has the required number of multiplicatively independent functional units.

Background

The paper explains that the main theorem applies to the cubic family with three linear factors, while its application to the analogous quintic family with five parameters remains conditional. The unresolved issue is specifically part (1) of Assumption 1.1, which requires sufficiently many multiplicatively independent units in the relevant function field; the authors also mention that the setting may include an additional variable z.