Is there a conditional conjugate prior distribution for the dispersion parameter of the negative binomial distribution given the mean rate parameter, if I parameterize it in this manner $$f(y_i; \mu, \kappa) := \frac{ \Gamma(y_i+1/\kappa) }{ \Gamma(y_i+1) \Gamma(1/\kappa) } \frac{(\kappa \mu)^{y_i}}{ (1+\kappa \mu)^{y_i+1/\kappa}}$$ for i.i.d. observations $Y_i \sim \text{NegBin}(\mu, \kappa)$ - i.e. the distribution is paramterized in terms of a mean rate parameter $\mu>0$ and a dispersion parameter $\kappa>0$ (with the distribution going towards a Poisson as $\kappa \rightarrow 0$)? I am ideally looking for a continuous conditionally conjugate density - rather than, say, a solution with point masses on a number of values for $\kappa$.
I have a scenario, where such a prior (or similarly a conjugate for something like $\log \kappa$ or any other convenient transformation) would be really useful, because I can easily obtain independent samples for $\log \mu$ and a conditional conjugate prior (or a mixture of such priors to be more flexible in expressing priors) would then make it very easy to also sample the dispersion parameter (and thus the full posterior).
It does not look to me like the answer is as simple as it's a gamma distribution - or at least by inspection of the likelihood I am failing to see it, but perhaps I am missing something obvious. Most of the information I managed to find was on different parameterizations that inconveniently (at least to me) do a change of variables (if you start out with the above parameterization) and mix up the mean rate and dispersion parameter of the parameterization above into a new parameter.