This place only introduces encounters with subordinates! If you're looking for encounters with subordinates, then it's definitely encounters with subordinates!
(Note: The phrase "下付き出会い" is somewhat ambiguous and context-dependent. Depending on the specific context, the translation might need adjustment. Please confirm the intended meaning.)
If this is about a specific type of meeting or encounter in a hierarchical context (like in workplaces), the above translation should work. However, if it involves a different nuance or cultural reference, further clarification may be needed.