Small-degree complete-intersection conjecture for quadratic manifolds

Let $X\subset {\mathbb P}^N$ be a quadratic manifold of dimension $n$. Small-degree complete-intersection conjecture. If

$$n\geqslant \operatorname{degree}(X)+1,$$

then $X$ is a complete intersection, unless it is projectively equivalent to

$$\mathbb G(1,4)\subset {\mathbb P}^9.$$

The source presents this as the optimal consequence expected from the Fano version of Hartshorne's conjecture. It also notes that small-degree manifolds are known to be complete intersections, but does not resolve the stated general claim.