I found (with some hints from a nice math.se user) numerically that $$\int_{S^{n}} x^2 dS = \frac{1}{n+1} \int_{S^{n}} dS$$ where $S^n$ is the unit $n$-sphere in $\mathbb{R}^{n+1}$ and $x$ of course the first dimension. Now one could use the unit $n$-sphere surface area formula for the RHS and try to solve the LHS integral in order to prove this, but I am only mildly interested in such an approach.
My actual questions are:
Is there some fancy trick that elegantly proves the conjecture without heavy integration, gamma functions, et cetera?
Can you provide some intuition on this remarkable factor $n+1$ holding in $\mathbb{R}^{n+1}$ ?
An exemplary consequence of the conjecture (and my gateway to this topic) is the covariance matrix $$\text{E}(\mathbf{uu}^\text{T}) = \frac{1}{n} \, \mathbf{I}_n$$ of uniformly random unit vectors $\mathbf{u} \in \mathbb{R}^{n}$.
By symmetry, $\int_{S^n}f(x_i)dS = \int_{S^n}f(x_j)dS$ for all $i,j$. Hence $$\begin{align}(n+1)\int_{S^n} x_0^2\ dS &= \int_{S^n}x_0^2\ dS + \int_{S^n}x_1^2\ dS + \ldots +\int_{S^n}x_n^2\ dS\\&=\int_{S^n}x_0^2 + x_1^2 + \ldots + x_n^2\ dS\\&=\int_{S^n} 1\ dS\end{align}$$