Bayer–Macrì–Stellari Castelnuovo inequality for curves on complete intersection threefolds

Let $X=Z(f_1,\dots,f_{n-3})\subset\mathbb{P}^n$ be a complete intersection threefold, where $f_i$ has degree $k_i$, and let $C\subset X$ be a curve of degree $d$ and genus $g(C)$. Bayer–Macrì–Stellari conjecture. One has

$$g(C)\leq\frac{2d^2}{3k_1\cdots k_{n-3}}+\left(\frac{5+3(k_1+\cdots+k_{n-3}-n-1)}{6}\right)d+1.$$

This conjecture generalizes the preceding low-degree genus bound to curves of arbitrary degree on complete intersection threefolds. It is motivated by the generalized Bogomolov–Gieseker inequality and remains open in the source context.