Let $d$ be a large positive integer. Let $f:mathbb R^d to mathbb R$ be a continuously differentiable function. Let $X$ be uniformly distributed on the unit-sphere $S_{d-1} := {x in mathbb R^d |x| = 1}$ and let $Z$ be a random vector in $mathbb R^d$ with iid coordinates from $N(0,1/d)$. Let $nabla_{S_{d-1}}f:S_{d-1} to mathbb R^d$ be the spherical gradient of $f$, defined by $nabla_{S_{d-1}} f(x) :=(I_d-xx^top)nabla f(x)$, where $nabla f(x)$ is the usual / euclidean gradient of $f$ at $x$.

Question.Is there any comparison inequality between $mathbb E|nabla_{S_{d-1}} f(X)|^2$ and $mathbb E|nabla f(Z)|^2$, meaning the existence of abolute constants $c,C>0$ independent of $f$ and $d$, such that $c mathbb E|nabla_{S_{d-1}} f(X)|^2 le mathbb E|nabla f(Z)|^2 le C mathbb E|nabla_{S_{d-1}} f(X)|^2$?