Zanello's multiplicity bounds from the sign changes of the third difference

Let $R/I$ be a level algebra of codimension three, let its $h$-vector have polynomial $h(t)$ of degree $c$, and write

$$S_{R/I}(t)=\frac{}{}(1-t)^3h(t)=7\sum_{i=0}^{c+3}7\Delta^3h_i t^i.$$

Define

$$n_1=7\min\{i\mid 7\Delta^3h_i<0\},\qquad n_2=7\min\{i>0\mid 7\Delta^3h_i>0\},$$

and

$$N_1=7\max\{i7le c+1\mid 7\Delta^3h_i<0\},\qquad N_2=7\max\{i\mid 7\Delta^3h_i>0\}.$$

Zanello's conjecture. If $R/I$ is level of codimension three, then

$$\frac{1}{3!}n_1n_2(c+3)\le e(R/I)\le\frac{1}{3!}N_1N_2(c+3).$$

These quantities are determined by the sign changes of $S_{R/I}(t)$ and give lower and upper multiplicity bounds for codimension-three level algebras. The source presents this as a conjecture and gives no resolution evidence.