The cyclotomic-number conjecture for conflict-avoiding codes

Let $p$ be an odd prime and set

$$\ell=[\mathbb{Z}_p^{\times}:\langle -1,2\rangle].$$

Let $A(i,j)$ denote the cyclotomic numbers of order $\ell$ associated with the subgroup $\langle -1,2\rangle$.

Cyclotomic-number conjecture. If $\ell\geq 3$, then

$$A(i,2i)\neq 0$$

for some $1\leq i\leq \ell-1$ with $(i,\ell)=1$.

The source states that this is equivalent to the quotient-group conjecture above, so it is another formulation of the same underlying claim rather than an independent conjecture. It is computationally verified for primes $p\leq 2^{30}$ in the supplied context, but remains unproved in general.