Consider $z, \tilde z\in \mathbb{S}^N$ on the hypersphere (i.e. $||z||_2 = 1$) and a unimodal conditional distribution $p(\tilde z|z)$ which is peaked at $z$ with $\int d\tilde z\,p(\tilde z|z)\,\tilde z \propto z$ and is symmetric (i.e. $p(\tilde z|z) = p(z| \tilde z)$).
For differentiable functions $h: \mathbb{S}^N\to\mathbb{S}^N$ I aim to understand the minima of the following quadratic functional:
\begin{align} \mathcal{L}[h] &\equiv \int_{\mathbb{S}^N} dz\,\ell_z[h], \\ &= \int_{\mathbb{S}^N} dz\,h(z)^\top\left[\int_{\mathbb{S}^N} d\tilde z\,(p(\tilde z) - p(\tilde z|z))\,h(\tilde z)\right], \end{align}
where $p(\tilde z) = 1/|\mathbb{S}^N|$ is a uniform distribution. The minima for each subexpression $\ell_z[h]$ are
$$ \min_{h}\ell_z[h] = \left\{\begin{array}{lr} -z, & \text{for } p(\tilde z) - p(\tilde z|z) \geq 0 \\ z, & \text{for } p(\tilde z) - p(\tilde z|z) < 0 \end{array}\right.. $$
In numerical evaluations (using Gaussian conditionals projected on the sphere) I find that the identity $h_{id}(z) = z$ is a global minimum, meaning that
$$ \forall h\quad \mathcal{L}[h_{id}] \leq \mathcal{L}[h]. $$
Looking at the minima of $\ell_z[h]$ and the symmetry of the problem that intuitively makes sense. It is also straight-forward to see that $h_{id}$ is at least a local extremum of $\mathcal{L}[h]$ (see remark 1 below).
Question: For which $p(\tilde z|z)$ is it possible to prove that the identity $h_{id}$ is a global minimum of $\mathcal{L}[h]$?
Remark 1 (Identity is local extrememum). To show that $h_{id}$ is a local extremum, it suffices to show that the first functional derivative of $\mathcal{L}$ vanishes in all directions $\eta: \mathbb{S}^N\to\mathbb{R}^N$,
$$ \left[\frac{d}{d\epsilon}\mathcal{L}[h + \epsilon\eta]\right]_{\epsilon = 0} = 2\lambda_h \int_{\mathbb{S}^N} dz\,\eta(z)^\top\int_{\mathbb{S}^N} d\tilde z\, (p(\tilde z)- p(\tilde z|z)) h(\tilde z) = 0. $$
To ensure that $(h + \epsilon \eta)(z)\in \mathbb{S}^N$ for small $\epsilon$ we need that $\eta(z)^\top h(z) = 0$ for all $z\in \mathbb{S}^N$, and hence all extrema $h^{*}$ of $\mathcal{L}$ must fulfill,
$$ \int_{\mathbb{S}^N} d\tilde z\, (p(\tilde z)- p(\tilde z|z)) h^{*}(\tilde z) \propto h^{*}(z). $$
Given that $p(z|\tilde z)$ fulfills $\int dz\,p(z|\tilde z)\,z \propto \tilde z$, it is straight-forward to see that $h_{id}$ fulfills this condition and is thus a local extremum.
Remark 2 (Additional constraints on $p(\tilde z|z)$). It's ok to add additional constraints on the conditional distribution if necessary (but the fewer constraints the better). Suitable additional constraints could be reflection symmetry (i.e. $p(-\tilde z|-z) = p(\tilde z|z)$) or rotation symmetry (i.e. $p(R\tilde z|Rz) = p(\tilde z|z)$ for any rotation $R$).