On smooth square-free numbers in arithmetic progressions

A. Booker and C. Pomerance (2017) have shown that any residue class modulo a prime $p\ge 11$ can be represented by a positive $p$-smooth square-free integer $s = p^{O(\log p)}$ with all prime factors up to $p$ and conjectured that in fact one can find such $s$ with $s = p^{O(1)}$. Using bounds on double Kloosterman sums due to M.~Z.~Garaev (2010) we prove this conjecture in a stronger form $s \le p^{3/2 + o(1)}$. Furthermore, using some additional arguments we show that for almost all primes $p$ one can replace $3/2$ with $4/3$.

We also consider more general versions of this question replacing $p$-smoothness of $s$ by the stronger condition of $p^{\alpha}$-smoothness.
Using bounds on multiplicative character sums and a sieve method, we also  show that we can represent all residue classes by a positive square-free integer $s\le p^{2+o(1)}$ which is $p^{1/(4e^{1/2})+o(1)}$-smooth.