I also found that using "cultural" and "competence" as key words in the contexts of English education or foreign language education for searching for articles using the university's database, I have come upon several relevant articles. One of them seems to be raising the same questions and approach that I would like to do, but I've only read the abstract so far.
While in the midst of coursework, I have found it difficult to reserve time to read for my dissertation. When I finally have some time to read, I find myself drained from reading materials for courses and writing papers. I need to find a way to prioritize my reading. It is not necessary for me to finish reading some papers for my research this semester, but it is important for me to finish reading for classes if I'd like a satisfactory grade. My strategy so far has been to skim over class readings that seem not to overlap with my research area. This strategy is mildly successful, but not successful enough for me to get the feeling of accomplishment.