What symbol would I use if I wanted to express that, in the context of some binary relation $P$ implied from context, that $\exists (a,b)\in P: a\ne b$, but not to the extent that $\forall (a,b) \in P: a\ne b$.

The use of this would be if one were discussing a more restricted system, but then move to discussing a less restricted one. Like, "if we know for sure that $a\cdot b=b\cdot a$, then .... However, if $a\cdot b \mathrel{\rlap{=}\,?} b\cdot a$, then the previous reasoning doesn't apply, so ...". ("$\mathrel{\rlap{=}\,?}$" instead replaced with the real symbol)