de Weger's conjecture on binomial collisions

Let $(n,k,m,l)$ be a binomial collision, meaning that $2\leq k\leq n/2$, $2\leq l\leq m/2$, $k<l$, and

$$\binom{n}{k}=\binom{m}{l}.$$

The known examples include the double collision, the six listed sporadic examples, and Lind's Fibonacci family.

de Weger's conjecture. There are no other collisions than those given above.

This conjecture asserts the completeness of the known classification of equal binomial coefficients under the stated ordering and size conditions. The source reports computational searches but gives no resolution of the conjecture.