Let $X\subseteq\mathbb P^n$ be a complex, projective variety. A linear subspace $L\subseteq\mathbb P^n$ will be called a maximal linear subspace of $X$ if $L\subseteq X$ and for any linear subspace $L'$ with $L\subseteq L'\subseteq X$, we have $L=L'$.
Let us denote by $$F_k(X):=\{ L\subseteq X \text{ linear subspace}, \dim(L)=k\}$$ the Fano variety of $k$-planes in $X$, which is a subariety of the Grassmannian of $k$-planes in $\mathbb P^n$. We set $F(X):=F_1(X)\mathrel{\dot\cup}\cdots\mathrel{\dot\cup} F_n(X)$. Then, consider in $F(X)$ the set of linear subspaces of $X$ which are maximal in the above sense, i.e. inclusion-wise maximal, not of maximal dimension. Denote this set by $\tilde F(X)$.
My question is: Is $\tilde F(X)$ an open subvariety of $F(X)$? Does anyone have an easy proof or a reference, in case it is?