Middle-coordinate abundance conjecture for bounded ordered integer simplices

Let $N$ be a natural number greater than $2$, let $d$ be a natural number strictly less than $N/2-1$, and let $n\geq N$ be another natural number. For the simplex $S^N(n)$ considered in the paper, compare $l$-element subsets whose coordinate values in a specified position are distinct. Middle-coordinate abundance conjecture. For every sufficiently small cardinality $l$, there are more $l$-element subsets of $S^N(n)$ with distinct $x_{d+1}$ coordinates than there are $l$-element subsets with distinct $x_d$ coordinates. This conjecture proposes a general form of the observed phenomenon that middle dimensions of the order simplex are more ample than its outer dimensions; the paper establishes only several specific identities and does not provide a general tool for proving the claim.