Let $\sigma$ be the uniform measure on $\mathbb{S}^{d-1}\subset \mathbb{R}^d$. For any region $R\subset \mathbb{S}^{d-1}$, let $X_R$ be a random variable which is uniformly distributed across $R$. We have $$E(X_R)=\frac{1}{\sigma(R)}\int_R xd\sigma (x)$$
Define a simplex in $\mathbb{S}^{d-1}$ to be the intersection of $d$ hemispheres.
Is it true that if $R\subset S\subset \mathbb{S}^{d-1}$ are spherical simplices then $|E(X_S)|\le |E(X_R)|$?
The progress I’ve made so far is in the following claim.
Claim: If $T\subset \mathbb{S}^{d-1}$ with $G=E(X_T)/|E(X_T)|$ and $B$ is a small ball $B$ disjoint from $T$, then $|E(X_{T\cup B})|<|E(X_T)|$ iff $B$ is further (by dot product) from $G$ than the rest of $T$, on average. That is, for $b\in B$
$$G\cdot b<\frac{1}{\sigma(T)}\int_T G\cdot xd\sigma(x)$$
Proof of claim: Note that for positive numbers $a,b,c,d$ satisfying $a/b<c/d$, $$\frac{a}{b}<\frac{a+c}{b+d}< \frac{c}{d}$$ Also, when $B$ is small, $$\frac{E(X_{T\cup B})}{|E(X_{T\cup B})|}\approx \frac{E(X_T)}{|E(X_T)|}=:G$$ so that $|E(X_T)|=G\cdot E(X_T)$ and $|E(X_{T\cup B})|\approx G\cdot E(X_{T\cup B})$. Expanding these out, the statement $|E(X_{T\cup B})|<|E(X_T)|$ becomes \begin{align*}\frac{\int_T G\cdot x d\sigma(x)+\int_B G\cdot x d\sigma(x)}{\sigma(T)+\sigma(B)}<&\frac{\int_T G\cdot x d\sigma(x)}{\sigma(T)}\\ \frac{\int_B G\cdot xd\sigma(x)}{\sigma(B)}<&\frac{\int_T G\cdot x d\sigma(x)}{\sigma(T)}\\ G\cdot b <& \frac{\int_T G\cdot xd\sigma(x)}{\sigma(T)}\end{align*} All the steps are reversible, so we have both directions of iff. One direction still works for any small region $B$ disjoint from $T$, but the other direction fails as the magnitude $|\int_B xd\sigma(x)|/\sigma(B)$ can be significantly less than $1$.
Just take a skinny simplex on a high dimensional sphere. The (spherical) centroid will be close to the boundary. Add a point in one more dimension, close to the centroid. In high dimensions, that will mostly only add points that are still close to the centroid. From your claim, it should increase the magnitude.