  The Debarre-de Jong conjecture predicts that the Fano variety of lines on a
smooth Fano hypersurface in $\mathbb{P}^n$ is always of the expected dimension.
We generalize this conjecture to the case of Fano complete intersections and
prove that for a Fano complete intersection $X\subset \mathbb{P}^n$ of
hypersurfaces whose degrees sum to at most 7, the Fano variety of lines on $X$
has the expected dimension.
