Two-axis extremizer conjecture for the radius-$\sqrt{2}$ reverse Littlewood–Offord problem

For each sufficiently large integer $n$, let $v_1,\ldots,v_n$ be arbitrary unit vectors in $\mathbb{R}^2$, and let $\epsilon_1,\ldots,\epsilon_n$ be independent Rademacher random variables. For an integer $t$ with $t\leq n$, define $v_i'=(1,0)$ for $i\leq t$ and $v_i'=(0,1)$ for $i>t$. Two-axis extremizer conjecture. For all sufficiently large $n$, there exists some $t\leq n$ such that

$$\mathbb{P}[\\|\epsilon_1v_1+\cdots+\epsilon_nv_n\\|_2\leq\sqrt{2}]\geq\mathbb{P}[\\|\epsilon_1v_1'+\cdots+\epsilon_nv_n'\\|_2\leq\sqrt{2}].$$

This asks whether the smallest radius-$\sqrt{2}$ concentration probability is attained by a configuration consisting of copies of the two coordinate-axis vectors. The source presents it as an open stronger statement and notes that it would identify the relevant extremal configuration.