Inductive-limit and projected-limit conjecture for twisted q-characters

Let $W$ be the Weyl group, let w\frac{in W, fix i\frac{in I and a\frac{inmathbb{C}^*, and let $m$ be a monomial in the variables $Y_{j,b}$. Let $V_\infty^w(m)$ be the corresponding inductive-limit representation, let $bpsi_{i,a}$ be the relevant $l$-weight, and let $pi_{q,\infty}^w(m)$ be the projected limit. Inductive-limit conjecture. The following assertions should hold:

$$T_w(bpsi_{i,a}^{-1}m)pi_{q,\infty}^w(m)=\chi_q(V_\infty^w(m));$$

the projected limit should factor as

$$pi_{q,\infty}^w(m)=c\times a,$$

with $c$ in the completed constant ring and $a$ in $T_w(bpsi_{i,a}^{-1}m)\overline{\mathscr{A}}_w$; and every irreducible $w$-highest-weight module $L_w(bpsi)$ in $\mathcal{O}^w$ should have a $q$-character factoring as

$$\chi_q(L_w(bpsi))=c\times a,$$

where $c$ is its constant part and $a$ is its non-constant part. The source presents these as conjectural relations intended to connect inductive limits, projected limits, and q-characters in $\mathcal{O}^w$.