The greedy $L_2$ discrepancy conjecture for the sequence generated by Algorithm 1

Let $\mathcal{S}^*$ be the sequence generated by Algorithm 1, and let $\widetilde{\mathcal{V}}$ be the symmetrized van der Corput sequence in base $2$. Write $L_{2,N}(\mathcal{S})$ for the $L_2$ discrepancy of the first $N$ elements of a sequence $\mathcal{S}$.

Greedy $L_2$ discrepancy conjecture. The sequence $\mathcal{S}^*$ should satisfy

$$\limsup_{N\to\infty}\frac{L_{2,N}(\mathcal{S}^*)}{\sqrt{\log N}}<\limsup_{N\to\infty}\frac{L_{2,N}(\widetilde{\mathcal{V}})}{\sqrt{\log N}}\leq 0.319553\ldots.$$

Numerical experiments indicate that $\mathcal{S}^*$ generally has smaller $L_2$ discrepancy than the symmetrized van der Corput sequence, while the displayed upper bound for the latter is attributed to Faure. An explicit description of $\mathcal{S}^*$ and sharper discrepancy bounds remain desirable.