Relationship between my wife and mother... what do I do?

Hello, I'm posting this because I'm so stressed out. I don't know where to start, but I've lurked here before and wanted to seek some advice. I'm 41 and my wife is 38. We don't have kids because we're struggling to conceive. My mother worked as a vendor in a traditional market for a long time, and maybe because she raised three sons, her voice is louder and her accent/tone is harsher than most people's. She's also very blunt. Before we got married, when I first introduced my wife, the first incident happened. A misunderstanding occurred while my mother was asking about my wife's family. According to my wife, she felt like she was being interrogated. My wife's father was an alcoholic, so my mother-in-law divorced him and raised two daughters alone. There's also the fact that my mother-in-law is estranged from her siblings due to inheritance disputes (my wife's grandfather left everything to the uncle, so they went to court). They were discussing things like that... From my mother's perspective, she was just really curious and kept asking. But my wife felt like she was being cross-examined. Even after marriage, my mom kept bringing up the same family issues, which eventually led to my wife exploding. We had a massive fight that day about her potentially moving out. There have been a few more incidents since then. My parents visited our place once, and my wife prepared a meal. I didn't really notice anything, but my wife says that while I was away, she was constantly criticized—that the food was too salty, too sweet, asking if she even washed the ingredients properly. She says she never heard a single 'thank you' or 'good job.' She even said my mom smirked after tasting the soup... And when she was peeling fruit for dessert, she got scolded for peeling off too much of the skin. I honestly don't know if she's exaggerating or not. Other than that, she says she doesn't want to talk to my mom or try to get on her good side because my mom just nitpicks everything she tries to do. So whenever I try to arrange a meal together, she makes every excuse in the book. Lately, she just flat-out says she's uncomfortable and doesn't want to go. I'm stuck in the middle and don't know what to do. My wife says my parents are low-key looking down on her and her mother. She says they wouldn't act like this if they actually liked her. She also mentioned that since neither side really helped us out financially, why does my mom act like her family is so much better off than ours? lol. Last weekend, my wife had to drive somewhere far for an errand. Since I couldn't make it, I asked if she could give my mom a ride because she was going in the same direction, but she apologized and refused. She said she wanted to travel 'in a good mood.' Should I just let things be and stay quiet? Or is it my job to fix this? If the latter is the answer, what am I supposed to do? I'm so worried that my headaches have been getting worse lately. Please give me some advice.