Dimension-free irrelevance of low-influence directions for convex sets

Let $K \subseteq \mathbb{R}^n$ be symmetric and convex, and let $v \in \mathbb{S}^{n-1}$ satisfy $\operatorname{Inf}_v[K] \leq \varepsilon$. A symmetric convex set $L$ can be compared with $K$ using the projection onto the $(n-1)$-dimensional subspace orthogonal to $v$. Low-influence direction conjecture. There is a symmetric convex set $L$ depending only on that projection such that

$$\Pr_{x \sim \mathcal{N}(0,1)^n}[K(x) \neq L(x)] \leq \tau(\varepsilon),$$

where $\tau$ depends only on $\varepsilon$, is independent of $n$, and tends to $0$ as $\varepsilon \to 0$. This is an open question that would provide a dimension-free robust form of the paper's irrelevance statement for zero-influence directions.