Determine polynomiality without the nonvanishing condition

Determine whether the symmetric-power \(L\)-function \(L({\rm Sym}^{k}Kl_{n,m}/\mathbb{F}_q,T)\) is always a polynomial when \(d_k(n,m,p)\neq 0\), without assuming \(d_k(n,m,p)=0\).

Background

The main theorem proves that the symmetric-power $L$-function is a polynomial only under the restriction $d_k(n,m,p)=0$, where $d_k(n,m,p)$ counts certain vanishing combinations of $(mn+1)$-st roots of unity in characteristic $p$.

The authors explicitly leave unresolved whether polynomiality persists when this condition fails. Resolving this would extend the polynomiality result to the full range of parameters satisfying $p\nmid m(mn+1)$.