Linear subspaces of minimal codimension in hypersurfaces

Let $k$ be a perfect field and let $X\subset {\mathbb{P}}^N$ be a hypersurface of degree $d$ defined over $k$ and containing a linear subspace $L$ defined over an algebraic closure $\bar{k}$ with $$\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m}${{\mathbb P}^N}L=r$\mathrm{.}Weshowthat$X$containsalinearsubspace$L_0$definedover$k$with$$\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m}${{\mathbb P}^N}L\le dr$. We conjecture that the intersection of all linear subspaces (over $\bar{k}$) of minimal codimension $r$ contained in $X$, has codimension bounded above only in terms of $r$ and $d$. We prove this when either $d\le 3$ or $r\le 2$.