Minimum distance conjecture for complete intersection ideals

Let $I\subset S:=K[t_1,\ldots,t_s]$ be a complete intersection graded ideal of dimension $1$, generated by forms $f_1,\ldots,f_c$ with $c=s-1$, where $d_i=\deg(f_i)$ and $2\leq d_i\leq d_{i+1}$ for $i\geq 1$. Suppose that the associated primes of $I$ are generated by linear forms. Minimum distance conjecture. For integers $d$, $k$, and $\ell$ satisfying

$$1\leq d\leq \sum_{i=1}^{c}(d_i-1)-1,$$ $$0\leq k\leq c-1,\qquad d=\sum_{i=1}^{k}(d_i-1)+\ell,\qquad 1\leq \ell\leq d_{k+1}-1,$$

one has

$$\delta_I(d)\geq (d_{k+1}-\ell)d_{k+2}\cdots d_c.$$

The conjecture holds when an initial ideal of $I$ with respect to some monomial order is a complete intersection, but remains open in general; it is sharp for vanishing ideals in the range where the minimum distance exceeds $1$.