Oda's conjectures on integer decomposition properties of smooth polytopes

Let a lattice polytope be smooth if at every vertex it has exactly $n$ edges whose primitive edge vectors form a lattice basis. A lattice polytope $P$ has the integer decomposition property (IDP) if, for every integer $k\geq 1$, every lattice point of $kP$ is a sum of $k$ lattice points of $P$. Two lattice polytopes $P$ and $Q$ form an IDP pair if

$$(P+Q)\cap\mathbb{Z}^n=(P\cap\mathbb{Z}^n)+(Q\cap\mathbb{Z}^n).$$

Oda's conjectures. The following two statements are true: (1) every smooth polytope has the IDP; and (2) if $P,Q\subset\mathbb{R}^n$ are smooth polytopes and the normal fan of $P$ refines the normal fan of $Q$, then $P$ and $Q$ form an IDP pair.

These conjectures concern lattice-point decompositions and, equivalently, projective normality of the associated projective toric varieties. The source presents them as questions raised by Tadao Oda; no resolution status is supplied here.