Stability-barrier conjecture for Hermite–Volterra discretizations

Let $L$ and $R$ denote the numbers of left- and right-sided Hermite degrees of freedom in a semi-discretized Hermite–Volterra (HV) method for an advection equation. Stability-barrier conjecture. The semi-discretized HV method is stable if and only if

$$0\leq R<L<R+3,$$

except when $(L,R)=(3,0)$, in which case the method is stable. This conjecture identifies the precise stability barrier for the class of HV methods; the supplied text presents it after examples of methods with different stencils, but gives no resolution.