Hansen's parameter estimate for toric codes

Let $X=X(\Delta)$ be a complete toric variety of dimension $r$, let $G$ be a $T$-invariant Cartier divisor, let $P_G$ be the polytope associated to $G$, and let $M$ be the lattice associated to $X$. For the toric evaluation code $C_L=C_L(G,P,X)$, write $n$ for its length and $d$ for its minimum distance. Suppose $X$ is nonsingular and projective, and suppose there is an integer $N>1$ such that

$$2N\cdot \operatorname{vol}(P_G)\leq n\leq 2N^2\cdot \operatorname{vol}(P_G).$$

Hansen's conjecture. If $q$ is sufficiently large, then every $f\in L(G)=H^0(X,\operatorname{\mathcal O}(G))$ has at most $n$ zeros in $X(\mathbb{F}_q)$, and consequently

$$d\geq n-2N\cdot \operatorname{vol}(P_G).$$

The meaning of sufficiently large may depend on $X$, $P$, and $G$, but not on $f$. The estimate was conjecturally proposed for toric codes and verified computationally for many examples; its general validity is not resolved by the supplied text.