Joyner's conjecture 4.3 on parameters of toric codes

Let $\mathcal{C}(E,D,X)$ be the toric code associated to the $1$-cycle $E$, the $T$-invariant Cartier divisor $D$, and the toric variety $X$. Assume that $X$ is a nonsingular toric variety of dimension $r$, that

$$\psi_D(v)=\min_{u\in P_D\cap M}\langle u,v\rangle$$

is strictly convex, and that $\deg(C)>\deg(D^r)$. Joyner's conjecture 4.3. If $q$ is sufficiently large, then every $f\in\mathrm{H}^0(X,\mathcal{O}(D))$ has at most $n$ zeros among the rational points of $X$. Consequently,

$$k\geq\dim\mathrm{H}^0(X,\mathcal{O}(D))=\#(P_D\cap M),$$ $$d\geq n-r!\,\#(P_D\cap M).$$

Moreover, if $n>r!\,\#(P_D\cap M)$, then

$$\dim\mathrm{H}^0(X,\mathcal{O}(D))=\#(P_D\cap M).$$

The paper gives a counterexample for $q\geq8$, so the conjecture is refuted.