Ask HN: Career Advice on Engineering Manager vs. Product Manager

My background has been in Software Engineering (California, Fortune 500). And now I am thinking of moving into either Engineering Management or Product Management. I already have an offer to be an Engineering Manager. I am open to MBA for the Product Manager role. I have always wanted to move into either EM or PM role. Based on my experience, I think I would enjoy both of those roles equally. I want to learn more in EM / PM role and then aspire to be at VP/Director position in 5-7 years. 1. Does EM or PM have a higher likelihood of getting to a Director / VP level position? 2. Is an MBA a good investment for either role in the long-term (should be able to get into the top 10 MBA school)? Especially, given I already have a very strong technical background and have business certification from Stanford. 3. Overall, what path should I follow (Engineering Manager or Product Manager)? Thank you so much for reading and helping!