Also, its been observed that pieces colliding together can be thrown into a higher orbit. Or more accurately, a more elliptic orbit with a much higher apogee than it had before. That means a Kessler Syndrome event in a lower orbit could start a chain reaction in higher orbits.

https://orbitaldebris.jsc.nasa.gov/quarterly-news/pdfs/ODQNv7i3.pdf

Also immediately obvious was the high susceptibility of the fragment to solar radiation pressure, demonstrated by rapid and dramatic changes in its orbit. From an initial orbit of about 1365 km by 1445 km with an inclination of 74 degrees, the fragment’s perigee began to decrease while its apogee increased. Within four weeks the orbit had been perturbed into one of 750 km by 1895 km. At this point, atmospheric drag became the dominant factor, causing the object to reenter the atmosphere a little more than two weeks later on 3 June (see figure). Thus, the fragment existed for only 43 days, despite originating in an orbit from which decay normally requires thousands of years.

Good news is that they won't last very long in orbit doing since the perigee is lowered. However, having so many fragments all at once would probably be very bad.