Numerical improvement conjecture for four-cell stratified sampling

Let $\mathbf{v}=(v_1,v_2,v_3)$ satisfy $v_1<v_2<v_3$ with each $v_i\in[0,\sqrt{2}]$. Let $\mathbf{\Omega}_{\mathbf{v}}^{(4)}$ be the associated four-cell partition, let $\mathcal{P}_{\mathbf{\Omega}_{\mathbf{v}}^{(4)}}$ denote a stratified point set from this partition, and let $\mathcal{P}_{\mathrm{jit4}}$ denote the four-point classical jittered-sampling set. Numerical improvement conjecture. There exists such a vector $\mathbf{v}$ for which

$$\mathbb{E}\,\mathcal{L}_2^2\left(\mathcal{P}_{\mathbf{\Omega}_{\mathbf{v}}^{(4)}}\right)<\mathbb{E}\,\mathcal{L}_2^2\left(\mathcal{P}_{\mathrm{jit4}}\right)=0.01909\ldots.$$

The claim is based only on numerical evidence; the authors report a partition with an estimated expected discrepancy of $0.0188\ldots$, but no proof is provided.