What the three examples say

Three outcomes, and they are different in kind:

• at $p = 89$ the obstruction improves 10 to 4, and 4 is sharp;
• at $p = 31$ the reduction bound is already the sharp value 4, so there is nothing left to remove;
• at $p = 29$ on the quintic the obstruction fires, improving 5 to 3, but 3 is not sharp and no precision brings it to 1.

Only the third is a statement about the method rather than about an attempt.

So the open question is not whether some prime works, but whether one can prove that some prime works.

Does this always work?

No.

Is there a prime for which the bound will be tight?