Casas–Alvero conjecture for univariate polynomials

Let $K$ be a field, and let $f\in K[x]$ be a non-constant monic univariate polynomial of degree $n$. For $1\leq i\leq n-1$, write $H_i(f)$ for its $i$-th Hasse derivative, and call $f$ a Casas–Alvero polynomial if $f$ has a non-constant common factor with each $H_i(f)$. Casas–Alvero conjecture. Assume that $\operatorname{char} K=0$. If $f\in K[x]$ is a Casas–Alvero polynomial of degree $n$, then there exists $b\in K$ such that

$$f(x)=(x-b)^n.$$

The conjecture predicts that the common-factor condition with all Hasse derivatives forces every root of $f$ to coincide. The paper studies bad primes and gives bounds relevant to reductions of this conjecture, but the characteristic-zero assertion itself is presented here without a resolution.