Let $X$ be a Banach space with a centered Gaussian measure $\mu_0$. Let $E$ be the Cameron-Martin space of $X$. Let the respective norms be $\|\cdot \|_X$ and $\|\cdot \|_E$. It is well known (see prop 3.30 here https://arxiv.org/pdf/0907.4178.pdf) that

$$\|u\|_X\leq C\|u\|_E$$

for all $u\in X$. There is no reverse inequality in general because $E$ is a strictly smaller subset of $X$. But what about locally?

That is, consider the set $Y_Z:=\{u\in X: \|u\|_E<Z, \|u\|_X<Z\}$. Then is there some $C_Z$ so that we have the reverse inequality on $Y_Z$?