Non-concentration conjecture for normalized theta-coefficients

Let $E$ be a semistable elliptic curve over $\mathbb{Q}$ with conductor $N$, and for every real cyclic extension $F/\mathbb{Q}$ of degree $d\geq3$ and conductor $m$ define the normalized theta-coefficient

$$\widetilde c_{F,\gamma}=\frac{c_{F,\gamma}\sqrt d}{\sqrt{\varphi(m)\log(m)}}.$$

Let $\Sigma_{d,\alpha,\beta}(X)$ be the multiset of $\widetilde c_{F,\gamma}m^\alpha\log(m)^\beta$ for real cyclic degree-$d$ extensions with $m<X$ and generic or special$^+$ $\gamma$. Non-concentration conjecture. There is a constant $B_E>0$ and, for every $d\geq3$, real numbers $\alpha_d,\beta_d$ such that for every real open interval $(a,b)$,

$$\limsup_{X\to\infty}\frac{\#\bigl(\Sigma_{d,\alpha_d,\beta_d}(X)\cap(a,b)\bigr)}{\#\Sigma_{d,\alpha_d,\beta_d}(X)}<B_E(b-a),$$

while $\{\alpha_d\varphi(d):d\geq3\}$ is bounded and $\lim_{d\to\infty}\beta_d=0$. This is a deliberately weaker prediction than convergence to a nonzero normal distribution, motivated by computations and random-matrix heuristics.