Ma–Zha–She's quotient-group conjecture for conflict-avoiding codes

Let $p$ be an odd prime, let $\mathbb{Z}_p^{\times}$ be the multiplicative group of nonzero residues modulo $p$, and let $\langle -1,2\rangle$ be the subgroup generated by $-1$ and $2$. Suppose

$$[\mathbb{Z}_p^{\times}:\langle -1,2\rangle]\geq 3.$$

The quotient $\mathbb{Z}_p^{\times}/\langle -1,2\rangle$ is cyclic.

Ma–Zha–She's quotient-group conjecture. There is a generator $t\langle -1,2\rangle$ of $\mathbb{Z}_p^{\times}/\langle -1,2\rangle$, for some $t\in\mathbb{Z}_p^{\times}$, such that

$$1+b=c$$

for some $b\in t\langle -1,2\rangle$ and $c\in t^2\langle -1,2\rangle$.

This conjecture implies the preceding coset formulation by multiplying the relation by successive powers of $t^3$. The source reports computational verification for primes $p\leq 2^{30}$ but gives no general proof.