Let $v$ be a given vector with $\|v\|_{\Sigma^{-1}} \leq 1$, where $\Sigma$ is a positive semi-definite matrix and $\|v\|_{\Sigma^{-1}} = \sqrt{v^\top\Sigma v}$. Meanwhile, let $u$ be a random vector drawn from $N(0,\Sigma^{-1})$. We know that for any given vector $\phi$, it holds that 
$$
P(\phi^\top u > \phi^\top v) > c
$$
where $c$ is a given positive absolute constant that depends on $v$. 

Question: Does a similar anti-concentration property still hold when we additionally condition on the event 
$
\mathcal{E} = \{u \in \mathcal{C}\}
$, where $\mathcal{C}$ is a given set such that $v \in \mathcal{C}$ and moreover $v$ is in the interior of $\mathcal{C}$? In other words, does it hold that 
$$
P(\phi^\top u > \phi^\top v \,|\, \mathcal{E})> c',
$$
where $c'$ is a given positive absolute constant that depends on $v$ and $\mathcal{C}$? One particular example of $\mathcal{C}$ that I am interested in is 
$$
\mathcal{C} = \{u: \| u + w \| \leq 1 \},
$$
where $w$ is a given vector such that $\| v + w \| \leq 1-\delta$ with $\delta > 0$.