I am an assistant professor in a business discipline. A couple of months ago, I came across (through casual chat) that a senior colleague of mine plans to pursue a research idea that I myself was also planning to pursue. I asked whether I could join her and her co-author on the project and she declined. But she said I could work on it myself. We didn't share any details of our thoughts afterwards. So we don't know what exactly each other's research question and progress on the project.
I went ahead and explored the project with the research question I had in mind. Now I have a finished draft of my work. I don't know the status of my colleague's project.
Now I would like to submit my paper to a journal. But I am not sure whether I should publicize it by e.g. posting it online/on my website. I am afraid that my senior colleague (she will be on my tenure committee) will be pissed off if I do so. She may think that I am scooping her. On the other hand, without publicizing it, people in my field (or potential referees of my paper) will not know that this work is mine and the paper may not get exposure/comments (which may not be critical).
What should I do? I don't think I have done anything unethical, but somehow I feel that I am stuck in a dilemma.