Shares of oil companies such as BPCL, HPCL, and IOC have seen movement, with some reports indicating gains of up to 4% amid falling oil prices and fuel rate cuts, while others suggest a decline of up to 4% due to various factors including rising oil prices and geopolitical tensions. The reasons for the movement in share prices appear to be conflicting, with oil price fluctuations and global events potentially playing a role. The overall impact on oil stocks is unclear, with different outlets reporting different outcomes.