Short-interval upper-bound conjecture for k-full numbers

Let $k\ge 2$ be an integer, let $x\ge 1$ be real, and let $Q_k(t)$ denote the number of $k$-full numbers at most $t$. A number is $k$-full if every exponent in its prime factorization is at least $k$. Short-interval upper-bound conjecture. There exists a constant $C_k\ge 1$ such that

$$Q_k(x+y)-Q_k(x)\le C_k y^{1/k}$$

uniformly for $1\le y\le x$. This would give a uniform bound for the number of $k$-full numbers in short intervals, analogous to the bound for perfect $k$-th powers; its status is not resolved in the supplied text.