Bounded sequences having an even number of accumulation points

In their papers, Leonetti, Russo, Somaglia, Menet and Papathanasiou posed the question of whether there exists an infinite dimensional vector space of sequences in ${\mathrm{\ell }}_{\mathrm{\infty }}$ having (except for the zero sequence) an even amount of accumulation points. Here we answer this question in the negative, by showing that this previous set of sequences is not even $3$-lineable and, therefore, not lineable.