The van Mises Fisher (vMF) distribution ($p(x;\mu,\kappa) \propto e^{-\kappa\mu^\top x}$ can be considered a close equivalent of a symmetric Normal Distribution $\mathcal{N}(\mu, \mathbf{I})$ on the hypersphere. "Close" here means that the vMF distribution is spherically symmetric, has a similar bell-shape as the Gaussian distribution and is relatively easy to work with.
What is the closest equivalent of an axis-aligned elliptical Gaussian distribution $\mathcal{N}(\mu, \Sigma)$ (with diagonal covariance $\Sigma$) on a hypersphere?
I first thought it might be something like $p(z;\mu,\Sigma)\propto e^{z^\top\Sigma\mu}$ (as the direct extension of a vMF distribution), but that's equivalent to a standard vMF with a shifted mean $\Sigma\mu/\left\|\Sigma\mu\right\|$.