There's a mountain of research. I'll cite the classic pro-homework paper by Cooper which synthesizes a whole bunch of moderate to low quality studies. https://emilkirkegaard.dk/en/wp-content/uploads/Does-Homewor... , because I think it shows how lacking the optimistic case is and lets you get to a whole lot of the rest of the research both ways.

A small (effectively zero) positive correlation was found between the amount of time a student says they spent on homework in elementary and achievement, but a slight negative (effectively zero) correlation was found when looking at the time parents reported students spending on it.

These measures don't show a causal relationship: "good students" are more likely to do homework. "poor students" are more likely to take a lot of time to complete the same amount of homework.

Elementary student reports of time spent on homework are not reliable. Parent reports are even worse.

The only causal evidence of a benefit in assigning homework in elementary comes from two short-term intervention studies: whether giving a homework take-home worksheet about vocabulary improves performance on the next vocabulary test (it seems so, very slightly). The longest term observational studies show a negative effect.