They will keep the power on, but only by wasting power. That won't help with your 3-4 week target.
Really bad! I've used wemos D1 mini (esp8266 based) in battery projects and they consume around 150~200uA in deep sleep, which I thought as not great, but compared to 9mA...
I would look for a better esp32 board, one more suited to battery powered projects, that can achieve < 0.5 mA in deep sleep. Use a 18650 size li-ion battery or a ~2000mAh li-po battery and you should hit 3-4 weeks life, I think. Some esp32 boards contain charging circuit for these types of battery.