We present the concept of an ultracold neutron (UCN) source with a superfluid He-4 (SF ⁴He) converter located in the thermal column of the WWR-K research reactor at the Institute of Nuclear Physics (INP) in Almaty, Kazakhstan. The conceptual design is based on the proposal of accumulating UCNs in the source and effectively transporting them to experimental setups. We propose to improve the UCN density in the source by separating the heat and UCN transport from the production volume and decreasing the temperature of the SF ⁴He converter to below about 1 K. To obtain operation temperatures below 1 K, we plan to use a He-3 pumping cryogenic system and minimize the thermal load on the UCN accumulation trap walls. Additional gain in the total number of accumulated UCNs can be achieved through the use of a material with a high critical velocity for the walls of the accumulation trap. The implementation of such a design critically depends on the availability of materials with specific UCN and cryogenic properties. This paper describes the conceptual design of the source, discusses its implementation methods and material requirements, and plans for material testing studies.