The conjecture that squarefree numbers have arbitrarily small power-sized gaps

A positive integer is squarefree if it is not divisible by the square of any prime. For $x>0$ and $\epsilon>0$, consider the short interval

$$(x,x+x^\epsilon].$$

Squarefree-gap conjecture. For any $\epsilon>0$ and $x$ sufficiently large in terms of $\epsilon$, the interval $(x,x+x^\epsilon]$ always contains a squarefree number.

Squarefree numbers have positive density, and this conjecture asserts that their gaps are smaller than every fixed positive power of $x$. The paper proves the weaker bound with interval length $Cx^{5/26}$ for some constant $C>0$, so the conjecture remains open.