On a conjecture about cellular characters for the complex reflection group $G(d,1,n)$

We propose a conjecture relating two different sets of characters for the complex reflection group $G(d,1,n)$. From one side, the characters are afforded by Calogero-Moser cells, a conjectural generalisation of Kazhdan-Lusztig cells for a complex reflection group. From the other side, the characters arise from a level $d$ irreducible integrable representations of ${\mathcal{U}}_q({\mathfrak{s}\mathfrak{l}}_{\mathrm{\infty }})$. We prove this conjecture in some cases: in full generality for $G(d,1,2)$ and for generic parameters for $G(d,1,n)$.