Nelson's quadratic Hecke-sum conjecture

Let $f\in H_k(R(1))$, let $\psi\in C_0^\infty(0,\infty)$, and let $q\in\operatorname{\mathbb{Q}}[x]$ be an irreducible quadratic integer-valued polynomial. Nelson's quadratic Hecke-sum conjecture. One has

$$\lim_{k\to\infty}\frac{\sum_{n\geq 1}\lambda_f(\lvert q(n)\rvert)\psi\!\left(\frac{n}{k}\right)}{kL(1,\operatorname{Sym}^2(f))}=0.$$

This conjecture is one of the arithmetic inputs used in Nelson's approach to mass equidistribution and quantum unique ergodicity. The paper uses the fixed-polynomial and fixed-test-function version displayed here, while noting that Nelson's formulation also varies them with $k$; its resolution is not stated in the supplied text.