The sharp writhe conjecture for smoothly hard max-tb Legendrian unknots

Let $\mathrm{minwr}_{\mathbb{R}^2}$ and $\mathrm{minwr}_{S^2}$ denote the minimum writhe of a max-tb Legendrian unknot front projection that is smoothly hard on the plane and sphere, respectively. Sharp writhe conjecture. For smoothed max-tb Legendrian unknots,

$$\mathrm{minwr}_{\mathbb{R}^2}=3,\qquad \mathrm{minwr}_{S^2}=4.$$

This conjecture sharpens the bounds established in the paper, namely $2\leq \mathrm{minwr}_{\mathbb{R}^2}\leq 3$ and $2\leq \mathrm{minwr}_{S^2}\leq 4$, and would give a sharp writhe obstruction for smoothly spherically hard max-tb unknot fronts.