The higher-rank torus-knot tail conjecture for $p<r$

Let $2\leq p<p'$ be coprime positive integers, let $2\leq r$ with $p<r$, and let $T(p,p')$ be the $(p,p')$ torus knot. Let $L_r(n\Lambda_1)$ denote the irreducible $\mathfrak{sl}_r$-representation of highest weight $n\Lambda_1$, let $\widehat{J}_{T(p,p')}$ be the normalized coloured invariant used in the paper, let $\delta_r$ be the Weyl vector of $\mathfrak{sl}_r$, and let $\Phi^+_0=\Phi^+_1=\emptyset$. The higher-rank torus-knot tail conjecture.

$$\lim_{n\rightarrow\infty}\widehat{J}_{T(p,p')}(L_r(n\Lambda_1))=\dfrac{\prod_{\alpha\in\Phi^{+}_{r-p}}(1-q^{(\alpha,\delta_{r-p})})}{\prod_{\alpha\in\Phi^{+}_{r}}(1-q^{(\alpha,\delta_{r})})}(q;q)^{p-1}_\infty\cdot\overline{\chi}^{p,p,p'}_{0,0}.$$

The conjecture extends the pattern observed in the cases $r=p$ and in computer experiments for $r=3,4,5$ with $p<r$; the supplied text does not establish it for the general range stated.