Dimension of the space of conics on Fano hypersurfaces

R. Beheshti showed that, for a smooth Fano hypersurface $X$ of degree $\leq 8$ over the complex number field $\mathbb{C}$, the dimension of the space of lines lying in $X$ is equal to the expected dimension. We study the space of conics on $X$. In this case, if $X$ contains some linear subvariety, then the dimension of the space can be larger than the expected dimension. In this paper, we show that, for a smooth Fano hypersurface $X$ of degree $\leq 6$ over $\mathbb{C}$, and for an irreducible component $R$ of the space of conics lying in $X$, if the $2$-plane spanned by a general conic of $R$ is not contained in $X$, then the dimension of $R$ is equal to the expected dimension.