The bounded h-star coefficient conjecture for lattice pyramids

Let $c$, $d$, and $i \in \{1,\ldots,d\}$ be fixed. A lattice polytope is a lattice pyramid if it is a lattice pyramid over a lower-dimensional lattice polytope. For an $n$-dimensional lattice polytope $P$, let $h^*_i$ denote the coefficient of $t^i$ in its $h^*$-polynomial, and let $\deg(P)$ denote its degree.

Lattice-pyramid conjecture. There is a function $f_i$ depending only on $c$ and $d$ such that any $n$-dimensional lattice polytope $P$ with $h^*_i=c$ and degree $\deg(P)=d$ is a lattice pyramid over an $(n-1)$-dimensional lattice polytope whenever

$$n \geq f_i(c,d).$$

For $i=1$, this follows from the paper's theorem. For $i=2,\ldots,d-1$, it would follow from the inequalities $h^*_1\leq h^*_i$, which are known in the case $d=n$ but have counterexamples for arbitrary degree. For $i=d$, the conjecture is equivalent to Batyrev's conjecture that $\operatorname{Vol}(P)$ is bounded by a function of $d$ and $h^*_d$.