Existence and uniqueness conjecture for specialized non-symmetric Macdonald polynomials

Let $m$ be a positive rational number, $p$ a natural number, and $[?]$ a composition such that ${\rm Coeff}_p[E_{\mu},m]$ is well defined and non-zero. For a composition $\nu$, define the specialization, when it exists, by

$$E_{\nu}(z;t^{-m},t):=\lim_{q\rightarrow t^{-m}}E_{\nu}(z;q,t).$$

Existence and uniqueness conjecture. There exists a unique composition $\nu$ for which this specialization is well defined and

$${\rm Coeff}_p[E_{\mu},m]\propto E_{\nu}(z;t^{-m},t).$$

This conjecture generalizes the corresponding theorem from a special case to arbitrary values of $p$. The paper states that it was not proved in full generality, although its subsequent results on duality functions are proved independently of it; it remains an important conceptual cornerstone of the work.