The conjecture on binomial near collisions with difference one

A binomial near collision is a quadruple $(n,k,m,l)$ with $2\leq k\leq n/2$, $2\leq l\leq m/2$, and

$$\binom{m}{l}-\binom{n}{k}=d>0,$$

with $\binom{m}{l}\geq d^3$. The listed examples in this section are the known near collisions with difference $d=1$.

Difference-one near-collision conjecture. There are no other near collisions with difference $1$ than those given above.

The table is complete for several specified pairs $(k,l)$, up to the stated bound $\binom{n}{k}\leq 10^{30}$, by reduction to integral points on elliptic curves. The global completeness assertion beyond those verified cases remains open in the source.