Generalized Lehmer conjecture for Hecke polynomial coefficients

Fix integers $m\geq 1$ and $r\geq 1$. Let $T_m(N,k)(x)$ be the Hecke polynomial, with its $r$-th coefficient understood in the source's indexing. Generalized Lehmer conjecture for Hecke polynomial coefficients. The $r$-th coefficient of $T_m(N,k)(x)$ is nonvanishing for every $N\geq 1$ coprime to $m$ and every even $k$ satisfying

$$k=12r\quad\text{or}\quad k\geq 12r+4.$$

This conjecture simultaneously extends the generalized Lehmer conjecture for Hecke traces and Clayton et al.'s level-one coefficient conjecture. The paper reports verification in all but finitely many cases, while the asserted uniform nonvanishing remains open.