Existence of any genuine noncongruence modular cusp form satisfying Deligne bound

Let $a_f(n)$ be the $n$-th Fourier coefficient of the $q$-expansion of a modular cusp form $f$ of weight $k$, modular for a finite-index noncongruence subgroup of $\mathrm{SL}(2,\mathbb{Z})$. Assume $f$ is not modular for any congruence subgroup. It appears wide open, in general, whether there exists $\epsilon > 0$ such that \begin{equation} \frac{|a_f(n)|}{n^{\frac{k-1+\epsilon}{2}}} \end{equation} is unbounded as $n \rightarrow \infty$, i.e., violating the Eichler-Shimura-Deligne bound. I have two questions, one concrete and one more open-ended.

1. Does there exist a single example of a genuine noncongruence modular form that satisfies the Eichler-Shimura-Deligne bound?
2. Deligne's deduction of the Ramanujan-Petersson conjecture from his purity theorem (see Deligne's proof of Ramanujan's conjecture for discussion about summarizing this) certainly relies on the Eichler-Shimura congruence relation, realizing the $p$-th Hecke operator $T_p$ as the sum of Frobenius and its transpose mod $p$. Of course, proving this congruence relation depends on understanding the reduction mod $p$ of the level $N$ modular curve $X(pN)$ for $p \nmid N$. In the general noncongruence case, understanding the mod $p$ reduction at bad primes of the connected components Hurwitz stacks is in general a hard problem, but one which appears approachable in some cases. On the other hand, while in some low genus cases (genus of the underlying noncongruence modular curve), Atkin Swinnerton-Dyer congruences + Rankin-Selberg bounds on $|a_f(n)|$ may allow one to relate the arithmetic of an integral model of the noncongruence curve to $a_f(n)$, this approach seems very far from yielding sharp bounds on $|a_f(n)|$. Here is my question; should one expect the Fourier coefficients of noncongruence cusp forms to have any direct relationship to an endomorphism acting on some 'cohomological object', in analogy to that between trace of Hecke operator and Frobenius eigenvalue on etale cohomology?