Erdős's odd-dimensionality-adjusted reverse Littlewood–Offord conjecture

Let $v_1,\ldots,v_n$ be unit vectors in $\mathbb{R}^2$, where $n$ is odd, and let $\epsilon_1,\ldots,\epsilon_n$ be independent Rademacher random variables. Odd-$n$ Erdős conjecture. There exists an absolute constant $c>0$ such that

$$\mathbb{P}[\\|\epsilon_1v_1+\cdots+\epsilon_nv_n\\|_2\leq 1]\geq \frac{c}{n}.$$

The paper proves the corresponding bound with radius $\sqrt{2}$ and explains that its method gives some radius strictly below $\sqrt{2}$ for odd $n$, but reaching radius $1$ appears to require new ideas. Thus this strengthened odd-$n$ statement remains open.