Let $X \subset \mathbb{P}^n$ be a non-singular complete intersection of $s$ hypersurfaces of degrees $d_1,\dots,d_s$ over an algebraically closed field $k$ of characteristic zero. Let $d=d_1 + \dots + d_s$. Suppose that $X$ is Fano, i.e. $d < n + 1$. Denote by $F_1(X)$ the Fano variety of lines in $X$. Then it is known that if $X$ is general, then one has
$$\dim F_1(X) = 2n - d - s - 2 \quad (*),$$
if this number is non-negative. Moreover, a conjecture of Debarre and de Jong states that $(*)$ holds for all such $X$ (not necessarily general) if $s=1$, i.e. if $X$ is a hypersurface. I have only seen this conjecture stated for hypersurfaces. My question is what happens for complete intersections.
Is $(*)$ still expected to hold for all such $X$ (not necessarily general) if $s >1$?
Perhaps this is dealt with in the original article where Debarre and de Jong made this conjecture, however I have been unable to find it, only other papers (not by Debarre and de Jong) which state this conjecture (without reference).