Weak and strong Eisenbud–Goto conjectures for projective toric varieties

Let $X\subseteq\mathbb{P}^r$ be a projective toric variety. Its maximal generator degree is denoted by $\operatorname{maxdeg} X$, and its regularity, degree, and codimension by $\operatorname{reg} X$, $\deg X$, and $\operatorname{codim} X$, respectively. Toric Eisenbud–Goto conjecture. Both inequalities should hold:

$$\operatorname{maxdeg} X\leq\deg X,$$ $$\operatorname{reg} X\leq\deg X-\operatorname{codim} X+1.$$

The first is called the weak inequality and the second the strong inequality. The paper states that the Eisenbud–Goto conjecture is widely believed for projective toric varieties, while even the weak form remains open.