Hecke lifting conjecture for framed links

Let $cmathcal{L}$ be a framed oriented link in $S^3$ with framing $cvec{ctau}=(ctau^1,\cdots,ctau^L)$. For a partition $cmu$, let $cZ_{cmu}(cmathcal{L};q,a)$ be the reformulated framed colored HOMFLY-PT invariant, let $ccheck{cZ}_{cmu}=cbrace{cmu}cZ_{cmu}$, and let $cPsi_p(f(q,a))=f(q^p,a^p)$ be the Adams operator. Write $Acequiv Bcmod C$ when $(A-B)/Ccincmathbb{Z}[z^2,a^{cpm1}]$, where $z=q-q^{-1}$ and $[p]=(q^p-q^{-p})/(q-q^{-1})$. Hecke lifting conjecture. For any prime number $p$,

$$\check{\mathcal{Z}}_{p}(\mathcal{L};q,a)\equiv (-1)^{(p-1)\sum_{\alpha=1}^L\tau^\alpha} \Psi_p(\check{\mathcal{Z}}(\mathcal{L};q,a)) \mod [p]^2.$$

This conjecture is a proposed integrality and congruence property for reformulated colored HOMFLY-PT invariants, motivated by the Chern-Simons/topological-string duality. The supplied text does not state whether this general framed-link version has been proved or disproved.