The upper bound conjecture for equivalence-class intersections in simplicial semigroup rings

Let $B$ be the affine semigroup under consideration, let $B_A$ denote its simplicial subsemigroup, and let $\sim$ be the equivalence relation on $B$. For $x,y\in B$, define

$$h(x,y):=(\min\{x_{[1]},y_{[1]}\},\min\{x_{[2]},y_{[2]}\},\ldots,\min\{x_{[d]},y_{[d]}\}).$$

For $x,y\in B\setminus\{0\}$, let $\delta(x,y)$ be the minimum, over the associated sequences, of the number of equivalence-class coincidences between the sequences for $x$ and $y$, minus two. Upper bound conjecture. Let $x,y\in B_A\setminus\{0\}$ with $x\sim y$. Then

$$\delta(x,y)\leq \deg h(x,y)-1.$$

The bound was proposed to control the number of equivalence classes arising from two sequences in the proof of the Eisenbud–Goto conjecture for monomial curves and simplicial semigroup rings. It is refuted: for equivalent elements $e_1,e_2\in B$, one has $\delta(e_1,e_2)=2>-1=\deg h(e_1,e_2)-1$.