Kumar Murty's conjecture on rational Hecke eigenvalues

Let $f$ be a weight $2$ normalized cuspidal Hecke eigenform of level $\Gamma_1(N)$ with quadratic coefficient field and without inner twists. Write $a_p(f)$ for its $p$-th Hecke coefficient. Kumar Murty's conjecture. There is a constant $c_f$ depending on $f$ such that

$$\#\{p<x\text{ prime}\mid a_p(f)\in\mathds{Q}\}\sim c_f\frac{\sqrt{x}}{\log x}.$$

This refines the known density-zero result for primes with rational coefficient and is part of a conjecture posed by Kumar Murty, building on work of S. Lang and H. Trotter. The source does not state a resolution.