Positive-relator quotient conjecture for reductions to amenable groups

Let $(G,P)$ be an ordered group. Let $p_1,p_2\nin P$ be incomparable, meaning $p_1\not\leq p_2$ and $p_1\not\geq p_2$, and let $N$ be the normal subgroup of $G$ generated by $p_1p_2^{-1}$. The quotient ordered group is $(G/N,P/N)$.

Positive-relator quotient conjecture.

1. If $(G,P)$ reduces to an amenable group, then $(G/N,P/N)$ reduces to an amenable group.
2. If $(G,P)$ strongly reduces to an amenable group, then $(G/N,P/N)$ strongly reduces to an amenable group.

The conjecture would establish closure of the classes of ordered groups reducing, or strongly reducing, to amenable groups under quotients by positive relators, with potential consequences for further group-theoretic constructions.