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)
It might be easiest to write out "however, if $a$ does not necessarily equal $b$" or "however, if $a$ doesn't have to equal $b$" ... upon a quick Google search, there doesn't seem to be a clear symbol for what you need, and it doesn't take that long to write it out.
Given that you won't be using this phrase as often as you would "there exists" or "for all" or "if and only if", it seems unnecessary to have a separate symbol for it.