I was able to prove this property:
$-G(x)x + B = \int_x^B G(k) dk$ has at most one solution $x\in [A,B]$ for all uniform distributions $g(x)$ (with c.d.f. $G(x)$), with support [A,B] and $B> A\geq 0$.
However, does this property hold for the more general case in which g(x) is unimodal on the support [A,B]. Perhaps there is also a geometric proof which I am just unable to see at the moment.
The integration by parts formula when G is a CDF on $[a,b]$ is $$ \int_x^B x dG(x) = B - xG(x) - \int_x^B G(x)dx. $$ And the left-hand side is zero only if the right-hand side is. But then you're asserting for all $x\in [a,b]$, that integral is zero. That doesn't hold for all $x \in [a,b]$ for a conventional cdf.