Reduced-form conjecture for SFLT2 solutions

Let $p$ be an odd prime, let $K=\mathbb Q(\zeta)$ be the $p$-th cyclotomic field, and consider a solution of the SFLT2 equation. Reduced-form conjecture. If the SFLT2 conjecture fails for $p$, then the solution(s) of the SFLT2 equation have the reduced form

$$u+\zeta v\in K^{\times p}.$$

The preceding discussion notes that the analogous reduced form is known when $p$ is regular, while the conjecture extends this assertion to all odd primes, including irregular ones.