As Ref.1 pointed out, aromatic nitrogen heterocycles in which the nitrogen contributes only one electron to the $\pi$-system are electrophilic as compared to benzene, and show a reduction by lithium aluminum hydride, an active reducing agent. Thus, nitrogen heterocycles are more electrophilic than benzene and are susceptible to attack by the hydride ion from a complex metal hydride anion (e.g., $\ce{NaBH4}$). However, in protic solvents, the intermediate cyclic enamines can undergo further reduction such as heterocyclic ring opening. For example, when $\ce{NaBH4}$ (1n $1:5$ excess) was used in refluxing 95% ethanol as the solvent to reduce heterocyclic quaternary salts such as OP's imidazolinium ion $(\bf{4})$, the intermediate cyclic enamine (in OP's case, it is $\bf{5}$) can further undergoes reduction to give ring opening product, a diamine within one hour (Ref.2). It is also possible to quickly generate various functionalized heterocycles including macrocyclic ones in addition to these highly substituted diamino derivatives.
Nonetheless, Shvydenko, et al. (2017) have published their result that described a way to reduce imidazolium salts without ring opening (Ref.3):
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/YcdaX.png)
Keep in mind that this reference was first mentioned here by Waylander in his comment to the question (many thanks). However, the authors have reported that all products having structure $\bf{5}$ (similar to OP's expected product) is unstable, specifically in even slight acetic condition (e.g., in NMR samples in $\ce{CDCl3}$). Bigger the fused ring, more the instability. This means open chain compounds without a fused ring such as OP's might be less stable, yet we don't know that for sure. Thus, it is tricky to use these conditions, but it is worth trying it to see the outcome.
References:
- Robert E. Lyle and Paul S. Anderson, "The Reduction of Nitrogen Heterocycles with Complex Metal Hydrides," Advances in Heterocyclic Chemistry 1966, 6, Pages 45-93 (DOI: https://doi.org/10.1016/S0065-2725(08)60575-9).
- Erik F. Godefroi, "Reductive ring cleavage of 1,3-disubstituted imidazolium iodides by sodium borohydride," J. Org. Chem. 1968, 33(2), 860-862 (DOI: https://doi.org/10.1021/jo01266a089).
- Tetiana Shvydenko, Kostiantyn Nazarenko, Kostiantyn Shvydenko, Sergey Boron, Oleksii Gutov, Andrey Tolmachev, Aleksandr Kostyuk, "Reduction of imidazolium salts – An approach to diazocines and diazocanes," Tetrahedron 2017, 73(49), 6942-6953 (DOI: https://doi.org/10.1016/j.tet.2017.10.053).