Crystal bases for reduced imaginary Verma modules of untwisted quantum affine algebras

We consider reduced imaginary Verma modules for the untwisted quantum affine algebras $U_q(\widehat{\text{g}})$ and define a crystal-like base which we call imaginary crystal base using the Kashiwara algebra ${\mathcal{K}}_q$ constructed in earlier work by Ben Cox and two of the authors. We prove the existence of the imaginary crystal base for any object in a suitable category $\text{mc}{O}_{red,im}^q$ containing the reduced imaginary Verma modules for $U_q(\widehat{\text{g}})$.