The product-order minimum conjecture for bidegrees of curves

Let $C/k$ be a curve, and let

$$S_{1,1}(C)=\left\{(a,b)\in\mathbb{N}^2\mid a\leq b\text{ and }C\simeq\text{ a curve of bidegree }(a,b)\text{ in }\mathbb{P}^1\times\mathbb{P}^1\right\},$$

where $\simeq$ denotes birational equivalence. Product-order minimum conjecture. The set $S_{1,1}(C)$ admits a minimum with respect to the product order $\leq\times\leq$ on $\mathbb{N}^2$. The conjecture concerns whether the bidegree invariant is independent of choosing a monomial order; no resolution is given here.