# Does sufficiency implies the reference measure to be a function of the sufficient statistic in exponential family?

Given the definition of exponential family $$f(x\mid\theta) = \exp\left( \eta(\theta)^\top T(x) + h(x) - A(\theta) \right)$$ for $\theta\in\Theta$ and $x\in\mathcal{X}$, doesn't this property of sufficient statistic $$T(x) = T(y), \forall x, y \in\mathcal{X} \Rightarrow f(x\mid\theta) = f(y\mid\theta), \forall \theta\in\Theta, \forall x, y \in\mathcal{X}$$ imply $h(x) = g(T(x)), \forall x\in\mathcal{X}$ for some $g$?