The Collatz Conjecture is a famous conjecture that has never been proven; nevertheless, there exists a simple heuristic probabilistic argument which supports its truth - in Wikipedia's words, "If one considers only the odd numbers in the sequence generated by the Collatz process, then each odd number is on average 3/4 of the previous one. (More precisely, the geometric mean of the ratios of outcomes is 3/4.) This yields a heuristic argument that every Hailstone sequence should decrease in the long run, although this is not evidence against other cycles, only against divergence."

Is there a similar heuristic probabilistic argument for the Navier-Stokes existence and smoothness conjecture, another problem about a dynamical system?

The Collatz Conjecture is a famous conjecture that has never been proven; nevertheless, there exists a simple heuristic probabilistic argument which supports its truth - in Wikipedia's words, "If one considers only the odd numbers in the sequence generated by the Collatz process, then each odd number is on average 3/4 of the previous one. (More precisely, the geometric mean of the ratios of outcomes is 3/4.) This yields a heuristic argument that every Hailstone sequence should decrease in the long run, although this is not evidence against other cycles, only against divergence."

Is there a similar heuristic probabilistic argument for the Navier-Stokes existence and smoothness conjecture, another problem about a dynamical system? In other words, is there a good reason to believe that this conjecture is true, even if we don't have a proof?