Finiteness conjecture for binomial near collisions of fixed difference

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 $\binom{m}{l}\geq d^3$. Here $d$ is the positive difference between the two binomial coefficients.

Fixed-difference finiteness conjecture. Given a fixed difference $d$, the number of near collisions with difference $d$ is finite.

This generalizes the preceding difference-one completeness conjecture from a classification claim to a finiteness claim for every fixed positive difference. The source provides no proof or resolution.