Asymptotic conjecture for the least smooth square-free representative

For a prime $p$, let $M(p)$ be the smallest integer such that every residue class modulo $p$ contains a $p$-smooth square-free representative.

Asymptotic conjecture for $M(p)$. As $p$ tends to infinity,

$$M(p)=p^{1+o(1)}.$$

The paper presents this as a stronger conjecture motivated by its bounds for $M(p)$ and by corresponding results for almost all primes. It remains open in the source.