A sublinear bound for the regularity of subspace arrangements

Let $X$ be the union of $k$ generic linear spaces in ${\text{PP}}^n$ of codimension at least $2$. We prove that the Castelnuovo-Mumford regularity of the vanishing ideal of $X$ grows in order not more than $\sqrt{k}$, and that this is an order-tight bound when the codimension equals $2$.