Conjecture on the vanishing probability of a fixed linking number in large grid models

Let $k$ be a fixed number, and consider links chosen randomly from links represented by grids whose size tends to infinity. Vanishing fixed-linking-number conjecture. The probability of randomly choosing a link with linking number $k$ converges to zero in the limit of infinitely large grids. In particular, almost all randomly chosen links in an infinitely large grid should be nontrivially linked. The conjecture concerns the concentration of random grid links away from every fixed linking number and would imply that nontrivial linking is typical in the large-grid limit. The supplied text gives no resolution of this conjecture.