Generalized Schur–Siegel–Smyth lower-bound conjecture for coefficient profiles

Let $\mathcal{F}$ be the class of monic, totally real algebraic-integer polynomials used in the Schur–Siegel–Smyth problem, written as

$$f(x)=\sum_{i=0}^n(-1)^{n-i}a_i x^i.$$

For $0<b<a$, define

$$h(a,b):=a\log a-b\log b-(a-b)\log(a-b),$$

and set

$$L(c):=e^{\frac{h(2-c,2-2c)-h(1,c)}{c}}.$$

Generalized Schur–Siegel–Smyth conjecture. Fix $\epsilon>0$. Then there exist only finitely many $f(x)\in\mathcal{F}$ such that

$$\left(\frac{a_{n-d}}{{n\choose d}}\right)^{1/d}\leq L(d/n)-\epsilon$$

for some $d$ with $0\leq d\leq n-1$.

This conjecture proposes a simultaneous lower bound for all normalized coefficients of polynomials in $\mathcal{F}$. The function $L(c)$ arises as the limiting coefficient profile from Siegel's construction, via Stirling's formula; the source does not state whether the conjecture has been resolved.