Exponential family — a complete sufficient statistic need not be unbiased for $\theta$ itself

Let $X_1,X_2,\dots,X_n$ be a random sample from the density $f(x,\theta)=c(\theta)h(x)e^{\pi(\theta)d(x)}, \theta\in\Theta$. Let $T(\underline x)=\sum_{i=1}^nd(x_i)$. Then which of the following statements is not true?