Conjecture on non-full-weight cyclic codes over $\mathbb F_3$

Let $n>3$ be a prime, and let $\operatorname{ord}_n(3)$ denote the multiplicative order of $3$ modulo $n$. Set

$$r=\frac{n-1}{\operatorname{ord}_n(3)}.$$

Let $\mathcal N(n,3)$ denote the family of cyclic codes over $\mathbb F_3$ containing no full-weight codeword. The cyclic-code counting conjecture. If $\operatorname{ord}_n(3)$ is odd, then

$$|\mathcal N(n,3)|=\sum_{j=0}^{\frac r2}\binom{r}{j}.$$

The formula refines a lower bound for $|\mathcal N(n,3)|$; the case $n=13$ gives $11$ cyclic codes, while the conjecture remains unproved in the stated generality.