= Solution
Take flat priors on $\alpha,\beta,\mu$ and scale priors $\pi(\sigma^2,\tau^2)\propto(\sigma^2\tau^2)^{-1}$ on the positive variances. The <posterior distribution> is then
$$
\pi(\alpha,\beta,\sigma^2,\mu,\tau^2\mid\mathcal D)
\propto\frac{L(\alpha,\beta,\sigma^2,\mu,\tau^2)}{\sigma^2\tau^2}.
$$
A random-walk <Metropolis–Hastings algorithm> can update $(\alpha,\beta,\mu,\log\sigma^2,\log\tau^2)$ with a symmetric proposal and accept a proposed state $z'$ from $z$ with probability $1\wedge\pi(z'\mid\mathcal D)/\pi(z\mid\mathcal D)$, including the Jacobian if the target is represented in transformed coordinates. Its transition kernel satisfies
$$
\pi(z)q(z,z')a(z,z')
=\min\{\pi(z)q(z,z'),\pi(z')q(z',z)\}
=\pi(z')q(z',z)a(z',z),
$$
which is <detailed balance>; hence the posterior is stationary.
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