Fano version of Hartshorne's complete-intersection conjecture

Let $X\subset {\mathbb P}^N$ be a smooth projective variety of dimension $n$ and codimension $c$ satisfying the hypotheses denoted by $(*)$ in the source. Fano Hartshorne conjecture. If

$$n\geqslant 2c+1$$

and $X$ is Fano, then $X$ is a complete intersection. This is stated as a weaker version of Hartshorne's conjecture. It is known when $c=2$, but remains open in general.