The 0–1 coefficient conjecture for fully supported representations

Let $G(\mathbb{R})$ be the real reductive group under consideration, with rank $l$, and let $\zeta_1,\ldots,\zeta_l$ be the relevant basis of $\mathfrak{h}_f^*$. For a fully supported representation in $\widehat{G(\mathbb R)}$, write its infinitesimal character as

$$\Lambda=\sum_{i=1}^{l}n_i\zeta_i,$$

where each $n_i$ is a nonnegative integer.

0–1 coefficient conjecture. One must have $n_i=0$ or $1$ for every $1\leq i\leq l$.

This speculation concerns the possible integral infinitesimal characters of fully supported representations. The source presents it as a conjectural pattern arising from calculations; no resolution is given here.