The odd-divisor classification conjecture for even nonsplitting perfect polynomials

Let $A$ be an even nonsplitting perfect polynomial over $\mathbb{F}_2$, and consider its odd divisors. A prime is called Mersenne or 2-Mersenne according to the terminology of the source. Odd-divisor classification conjecture. The only odd divisors of $A$ are Mersenne or 2-Mersenne. The source says that this conjecture, together with the preceding divisor-sum conjectures, would constrain the even nonsplitting perfect polynomials with the relevant prime-factor conditions; no proof or resolution is given.