Josephus Problem - Is there a position with 0 chance of surviving, regardless of any skip interval?

Question

Consider a variant of Josephus Problem. Instead of finding the initial position p_survive of the survivor, I want vary the skip interval k and determine if the following is true:

For all positions p in the circle, there exists a value of k such that p = p_survive exists.

Intuitively, it seems like the statement is true because the circle changes in size after each elimination while the skip interval remains constant, so there is a chance for each position to be the surviving position after using many values of k. However, I am not sure if there is a more robust proof (or disproof) for this.

Any help would be greatly appreciated, thanks!