A volume bound for lattice polytopes with trinomial $h^*$-polynomials

Let $\Delta$ be a lattice polytope whose $h^*$-polynomial is

$$1+a t^k+b t^{2k},$$

where $b\ge 2$. Volume-bound conjecture. One has

$$a+b+1\le (4b+4)k,$$

or equivalently,

$$\operatorname{vol}(\Delta)\le \frac{4b+4}{2}\deg(\Delta).$$

This conjecture extends the observed volume bound for the trinomial case with leading coefficient $b=1$, motivated by the general boundedness of lattice-polytope volume in terms of the degree and leading coefficient of the $h^*$-polynomial. The source presents it as a guess for a more general class of $h^*$-trinomials; its resolution is not specified.