Minimal-degree conjecture for non-degenerate integral curves in weighted projective three-space

Let $C$ be a non-degenerate integral curve in the weighted projective space $\mathbf{P}(1,1,m,n)$, where $m$ and $n$ are positive integers. Write $n\bmod m$ for the remainder on division of $n$ by $m$. Minimal-degree conjecture. Every such curve satisfies

$$\deg(C) \geq 1 + \frac{1}{m} + \frac{1}{n} - \frac{n \bmod m}{mn}.$$

The preceding bound is weaker by $1/n$ and may fail to be sharp when the associated morphism is not generically injective. The conjecture asserts the stronger lower bound suggested by the determinantal curve constructed in the paper, but no resolution is supplied here.