Logarithmic step bound for the large steps algorithm

Let $k\geq 3$ and let $\vec{\mathbf{v}}\in\mathbb{Z}^{k-1}\setminus\{\vec{\mathbf{0}}\}$. In the Large Steps Algorithm, let $j_{\mathrm{lsb}}$ denote the bound on the largest index used in the resulting decomposition. Large steps logarithmic-bound conjecture. There exist constants $c_k,d_k>0$ such that

$$j_{\mathrm{lsb}}\leq c_k\log\lVert\vec{\mathbf{v}}\rVert_2+d_k.$$

Consequently, $j_{\mathrm{lsb}}=O(\log\lVert\vec{\mathbf{v}}\rVert_2)=O(\log\lVert\vec{\mathbf{v}}\rVert_\infty)$. The paper proves this bound for $k=3$ and conjectures its extension to all $k\geq 3$ based on computational evidence; the corresponding general result remains open.