The second-moment lower-bound conjecture for lonely runners

Let $X$ be the random variable used in the paper to count the relevant loneliness events for a set of distinct speeds $v_1,\dots,v_n$, and let $\delta<1$. Second-moment lower-bound conjecture.

$$\mathbb{E}(X^2)\geq (1+o(1))4\delta^2 n^2+2\delta n.$$

The conjecture proposes an asymptotically sharp strengthening of the paper's lower bound for $\mathbb{E}(X^2)$. The authors explain that its proof would require controlling positive and negative error contributions; it is presented as an open question, while examples show that the second moment need not be $O(\delta^2n^2)$.