  We prove that the set of symplectic lattices in the Siegel space
$\mathfrak{h}_g$ whose systoles generate a subspace of dimension at least 3 in
$\mathbb{R}^{2g}$ does not contain any $\mathrm{Sp}(2g,\mathbb{Z})$-equivariant
deformation retract of $\mathfrak{h}_g$.
