Moving “virtual” singularity?

Imagine two close, really big black holes rapidly spinning around each other. That setup would emit a terrible amount of gravitational waves. My question is, could those gravitational waves, if big enough, temporarily collapse space around them into a singularity?

That is, not the black holes themselves forming a singularity, but the ripples they form on space-time, constructively interfering to form a singularity in a place where there is no actual mass.

The majority of the radiation produced isn't going to happen until the holes are orbiting very close to each other.${}^{1}$ The plunge process after the instability hits is also very rapid, creating a common horizon in much less time than, say, an orbit.
${}^{1}$For an extreme ratio binary pair of unspinning holes, the last stable orbit is at a radius of $\frac{6GM}{c^{2}}$, where $M$ is the mass of the larger hole. Radii of instability for extreme ratio spinning holes are typically closer than this.