Irreducibility of root-height generating polynomial

Question

The height $ht(\alpha)$ of a positive root $\alpha$ in a (finite, crystallographic) root system $\Phi$ is $\sum_{i=1}^n c_i$ where $\alpha = \sum_{i=1}^n c_i \alpha_i$ is its decomposition as a sum of simple roots. I am interested in the polynomial $$R_{\Phi}(x):=\sum_{\alpha \in \Phi^+} x^{ht(\alpha)-1}.$$

For example (using the usual Cartan-Killing nomenclature), we have:

• $R_{A_n}(x)=n+(n-1)x+\cdots +2x^{n-2}+x^{n-1}$
• $R_{B_n}(x)=R_{A_n}(x^2)+xR_{A_{n-1}}(x^2)$
• $R_{D_n}(x)=R_{B_n}(x)-\sum_{i=n-1}^{2n-2} x^i$

Questions:

1. When $\Phi$ is an irreducible root system, is $R_{\Phi}(x)$ an irreducible polynomial over $\mathbb{Q}$? This has been checked for the exceptional types ($G_2, F_4, E_6, E_7,$ and $E_8$) and for the infinite families listed above for $n \leq 500$.
2. If the answer to (1) is "yes", is there some uniform Lie-theoretic reason that this is true (that is, a proof not relying on the classification)?

Answer 1

(Turning Gjergji Zaimi's comment into a community wiki answer.)

A problem equivalent to the case of $R_{A_n}$ is discussed in Classes of polynomials having only one non-cyclotomic irreducible factor, by Borisov, Filaseta, Lam, and Trifonov (Acta Arithmetica 90 (1999), 121–153). They prove irreducibility in many special cases but the problem remains open (or at least, none of the papers that Google Scholar lists as citing this paper solves the problem).