Common-root conjecture for truncated binomials

For integers $d$ and $0<k<m<d$, let $P_{k;d}(x)$ and $P_{m;d}(x)$ denote the truncated binomial polynomials used in the paper. Common-root conjecture. The polynomials $P_{k;d}(x)$ and $P_{m;d}(x)$ do not have common roots. This is the additional condition needed for non-degeneracy in the case $N=2$; the source does not establish it for all $0<k<m<d$.