Conjecture on common roots of sorted binomial polynomials

Let $P_n(z)$ denote the sorted binomial polynomial, and call a root nontrivial when it is not the trivial root shared by the relevant polynomials. Common-root conjecture. $P_n(z)$ and $P_{n-k}(z)$ share no nontrivial roots for all $n$ and $1\leq k\leq n$. This generalizes the proved case $k=2$ discussed immediately beforehand; the claim is presented as a conjecture, and no resolution is given here.