The homogeneous-limit conjecture for eigenfunctions and generalized eigenfunctions

Let $\Delta=\{\alpha_1,\ldots,\alpha_{n-1}\}$ be the simple roots of $A_{n-1}$, let $Q_+$ be the positive cone of its root lattice, and let $C(\Delta)=\{x\mid (x,\alpha_i)\geq0,\ \alpha_i\in\Delta\}$. Let $f_\alpha$ denote the eigenfunction of $I(\alpha)$ or $D$ indexed by $\alpha\in Q_+$, with generalized eigenfunctions included when eigenvalues coalesce.

Homogeneous-limit conjecture. For $\alpha\in Q_+$, $f_\alpha$ remains finite at $s_1=\cdots=s_n=1$ if and only if $\alpha\in C(\Delta)$. Otherwise it diverges in this homogeneous limit and a generalized eigenfunction occurs. Moreover, in the homogeneous case every eigenspace is characterized by a Weyl-chamber element $\alpha\in Q_+\cap C(\Delta)$ and

$$V_{\lambda_\alpha}=\bigoplus_{\substack{\sigma\in W(A_{n-1})\sigma(\alpha)\in Q_+}}\mathbf{C}f_{\sigma(\alpha)}.$$

These statements describe the predicted Jordan-block and eigenspace structure when parameters specialize to the non-diagonalizable homogeneous point. They are inferred from explicit low-rank formulas and remain conjectural in general.