Conjecture on the merit factor of Gordon-Mills-Welch sequences

Let $F=\mathbb{F}_{2^d}$ and let $K$ be a subfield of size $2^k$, where $k$ divides $d$. Let $\theta$ be primitive in $F$, let $\ell$ be coprime to $2^k-1$, and let the Gordon-Mills-Welch sequence of length $n=2^d-1$ be the coefficient sequence of

$$\sum_{j=0}^{n-1}\psi\big(\operatorname{Tr}_{F/K}(\theta^j)^\ell\big)z^j.$$

Its negaperiodic and periodic versions are defined analogously. Gordon-Mills-Welch merit factor conjecture. For each such choice of $n$, $\theta$, $k$, and $\ell$, the asymptotic merit factor of the Gordon-Mills-Welch sequence, and of its negaperiodic and periodic versions, is the same as that of a Galois sequence as specified in the source's Theorem 2.2. This is motivated by numerical evidence; the asserted asymptotic equivalence remains unproved in the source.