Quadratic $h^*$-polynomial inequality

Let $\Delta$ be an $n$-dimensional lattice polytope whose $h^*$-polynomial is quadratic, written as

$$h_\Delta^*(t)=1+h_1^*t+h_2^*t^2.$$

Quadratic $h^*$-polynomial conjecture. One has

$$h_1^*\leq 3h_2^*+4.$$

This is proposed as a more precise statement for quadratic $h^*$-polynomials, refining the preceding volume-boundedness conjecture in that case. The source does not provide a resolution.