Ma–Zha–She coset equation conjecture

Let $p$ be an odd prime, let $H=\langle-1,2\rangle\subset\mathbb{F}_p^{\times}$, let $\ell_0$ be the index parameter used for $H$, and let $g$ be a generator of $\mathbb{F}_p^{\times}$. Ma–Zha–She conjecture. If $\ell_0\geq 3$, then there exist $b\in gH$ and $c\in g^2H$ such that

$$1+b+c=0$$

in $\mathbb{F}_p$. The conjecture gives the coset triples needed for the preceding construction and, unlike the preceding formulation, does not assume $4\nmid o_p(2)$.