Exponential family and conjugate priors Is a distribution that belongs to the exponential family necessarily conjugate prior?

A bit of reverse engineering: for the likelihood function associated with an exponential family, with natural parameterisation $\theta$ and sufficient statistic $x$
$$\ell(\theta|x)\propto \exp\{\theta^\prime x-\tau(\theta)\}\quad x\in\mathcal X,\ \theta\in\Theta$$
a family of conjugate priors is made of distributions with densities
$$\pi(\theta|x_0,\lambda)\propto \exp\{\theta^\prime x_0-\lambda\tau(\theta)\}\quad x_0\in\mathcal X_0,\ \lambda\in\Lambda$$
with respect to an arbitrary dominating measure $\text d\nu$ on $\Theta$, provided

*

*$\mathcal X_0+\mathcal X\subset\mathcal X_0$

*$\Lambda+\mathbb N \subset \Lambda$

*$\int_\Theta \exp\{\theta^\prime x_0-\lambda\tau(\theta)\}\,\text d\nu(\theta) <\infty$ for all $x_0\in\mathcal X_0,\ \lambda\in\Lambda$.

It clearly defines an exponential family with sufficient statistic $(\theta,\tau(\theta))$.
The reverse problem is to start from an exponential family density on $\theta$
$$\pi(\theta|\xi)\propto\exp\{T(\theta)^\prime\xi-\delta(\xi)\}\propto\exp\{T(\theta)^\prime\xi\}$$
and to identify a likelihood enjoying the above as a conjugate family. This means first breaking $T(\theta)^\prime\xi$ into $\theta^\prime x_0-\lambda\tau(\theta)$ and second finding  a measure $\text dμ$ on $\mathcal X$ such that
$$\int_{\mathcal X} \exp\{θ′x\}\text dμ(x)=\exp\{τ(θ)\}$$
This is an issue when considering for instance the Poisson distributions
$$\pi(\theta|\xi)\propto\exp\{\theta\xi-\log(\theta!)\}\mathbb I_\mathbb N(\theta)$$
because $\theta$ and $\xi$ are unidimensional. One would thus need to embed the Poisson family into a two-dimensional family
$$\pi(\theta|\xi,\lambda)\propto \exp\{\theta\xi-\lambda\log(\theta!)\}\mathbb I_\mathbb N(\theta)$$
and then check if there exist $\lambda_0\in\mathbb R$,  a set $\mathcal X\subset\mathbb R$, and a measure $\text d\mu$ on $\mathcal X$such that
$$\int_\mathcal X \exp\{\theta x\} \,\text d\mu(x)=\exp\{\lambda\log(\theta!)\}\quad \forall\theta\in\mathbb N$$
