Double lines in $mathbb{G}(k,n)$

In the Hilbert scheme $Hilb_2(mathbb{G}(k,n))$ of conics in $mathbb{G}(k,n)$ consider a point $(C)$ corresponding to a conic in $mathbb{G}(k,n)$ supported on a line and whose linear span is a plane not contained in $mathbb{G}(k,n)$.

Consider the union $bigcup_{Hin C}Hsubset mathbb{P}^n$ the union of all the $k$-planes parametrized by the conic $C$, and let $Lambda_Csubsetmathbb{P}^n$ be the linear span of $bigcup_{Hin C}H$. Is the dimension $k+1$ or $k+2$?