  Since the celebrated resolution of Kadison-Singer (via the Paving Conjecture)
by Marcus, Spielman, and Srivastava, much study has been devoted to further
understanding and generalizing the techniques of their proof. Specifically,
their barrier method was crucial to achieving the required polynomial root
bounds on the finite free convolution. But unfortunately this method required
individual analysis for each usage, and the existence of a larger encapsulating
framework is an important open question. In this paper, we make steps toward
such a framework by generalizing their root bound to all differential
operators. We further conjecture a large class of root bounds, the resolution
of which would require for more robust techniques. We further give an important
counterexample to a very natural multivariate version of their bound, which if
true would have implied tight bounds for the Paving Conjecture.
