The iterated-derivative toric residue conjecture

Let $A$ be a configuration, let $f$ be a rational $A$-hypergeometric function in variables $x_1,\dots,x_s$, and let a facial subset be a subset of $A$ arising as a face of the configuration. A toric residue is the residue construction associated with such configuration data. The iterated-derivative toric residue conjecture. Every rational $A$-hypergeometric function $f$ has an iterated derivative

$$\frac{\partial^{i_1+\cdots+i_s}}{\partial x_1^{i_1}\cdots\partial x_s^{i_s}}f$$

which is a toric residue defined by some facial subset of $A$. Families of examples and computer experiments support this conjecture, but the source gives no proof or resolution.