Stability of $\mathrm{ı}$canonical bases of locally finite type

We prove the stability conjecture of $\mathrm{ı}$canonical bases, which was raised by Huanchen Bao and Weiqiang Wang in 2016, for all locally finite types. To this end, we characterize the trivial module over the $\mathrm{ı}$quantum groups of such type at $q=\mathrm{\infty }$. This result can be seen as a very restrictive version of the $\mathrm{ı}$crystal base theory for locally finite types.