LLL candidate conjecture for modular symbols

Let ${\mathbf v}=[v_1,\dots,v_n]$ be a modular symbol with $\|{\mathbf v}\|>1$, and define

$$b({\mathbf v})=\sum_i v_i v_i^t.$$

Let $E$ be the standard basis of ${\mathbb Z}^n$. The quadratic form $b({\mathbf v})$ is called LLL-reduced when it is the Gram matrix of an LLL-reduced ordered basis. Let $\operatorname{cand}{\mathbf v}$ be the set of nonzero lattice points that can be written as $w=\sum_i q_i v_i$ with $0\leq |q_i|<1$. LLL candidate conjecture. If $b({\mathbf v})$ is an LLL-reduced quadratic form, then

$$\operatorname{cand}{\mathbf v}\cap E\ne\varnothing.$$

This conjecture offers an alternative, potentially efficient way to choose candidates for the modular symbol algorithm and is intended to give an explicit polynomial-time implementation. Its validity is presented as conjectural, with no resolution supplied in the paper.