He–Juškevičius–Narayanan–Spiro conjecture for odd planar Rademacher sums

Let $v_1,\dotsc,v_n\in\mathbb{R}^2$ be unit vectors, and let $\varepsilon_1,\dotsc,\varepsilon_n$ be independent random signs, each uniformly distributed in $\{-1,1\}$. He–Juškevičius–Narayanan–Spiro conjecture. There is a constant $c>0$ such that, for every odd $n$,

$$\mathbb{P}\!\left[\left\|\sum_{i=1}^n\varepsilon_i v_i\right\|_2\leq 1\right]\geq\frac{c}{n}.$$

This is the proposed odd-$n$ case of Erdős's conjecture, motivated by the fact that the known obstruction requiring radius $\sqrt{2}$ works only for even $n$; whether the asserted lower bound holds remains open in the supplied text.