Cyclotomic-class partition conjecture for products of primes congruent to $1$ modulo $6$

Let $n$ be a product of primes, each of which has the form $6k+1$. For $t>1$, let $\operatorname{Cos}(t)$ be the union of all binary $2$-cyclotomic classes of size $t$, and let $N_t$ be the number of those classes. For $a,b\in\mathbb{Z}_{2^n-1}\setminus\{0\}$, write $\operatorname{Cyc}\Gamma(a)$ for the union of the cyclotomic classes indexed by the elements of $\Gamma(a)$.

Cyclotomic partition conjecture. For every integer $t>1$, there are $N_t/18$ pairs $(a_i,b_i)$, $i=1,\ldots,N_t/18$, such that

$$\operatorname{Cos}(t)=\dot\bigcup_i\bigl(\operatorname{Cyc}\Gamma(a_i)\,\dot\cup\,\operatorname{Cyc}\Gamma(b_i)\,\dot\cup\,\operatorname{Cyc}\Gamma(a_i+b_i)\bigr).$$

Such a partition would group the relevant binary cyclotomic classes into triples of six-class configurations and is used to construct symmetric balanced triangle designs. The supplied text gives the assertion as a proposition-like statement but provides no resolution evidence.