What is a sensible choice for the parameter of a LKJ prior to get a reasonably vague prior in a hierarchical logistic regression?

For each of 70 individuals, there are daily yes/no outcomes (I would guess that probabilities very close to 0 or 1 would be rare) per day for 24 weeks and model the yes answer in a week as a binomial outcome. We assume a correlation across weeks by specifying a 24-d vector random effect per individual that is multivariate normal across individuals with a 24\times 24 correlation matrix. There is little known on the correlation over time (I’m pretty sure it’s not the identity matrix and probably the correlation drops off over time, but who knows…), so we’d like a prior that does not influence the posterior all that much and let the data to mostly determine the posterior correlation (it seems like we have a decent amount of data so that should hopefully be realistic).

Any thoughts or recommendations?