These are interesting findings, Elisa Lancini. I am not in the position to answer this authoritatively, so while waiting for an expert response, I would like to refer you to a resource I found that may be provide at least a tangential discussion of the issues you encountered: Article The Hartung-Knapp-Sidik-Jonkman method for random effects me...

Dear Aedrian Abrilla and those who are following the question,

I would like to summarise the answers I received by email after contacting some experts in the field.

-The intervals tend to become very conservative with few studies in the sample. When the standard errors are all equal, then the coverage of the HKSJ method is accurate. It deteriorates when the standard errors are unequal.

- It is implausible to associate the width of the interval with the heterogeneity of the standard errors.

- Thus the CI for the overall mean inherits its precision (CI width) from two factors: the precision of the studies entering the meta-analysis and the (empirical/observed) heterogeneity

So in the case of my plot and data, given that the sample size and standard errors within studies are similar between groups, we could interpret these results as due to heterogeneity in the effect sizes of the included studies.

Check https://bmcmedresmethodol.biomedcentral.com/articles/10.1186/s12874-015-0091-1

Also check https://bmcmedresmethodol.biomedcentral.com/articles/10.1186/s12874-018-0618-3

Elisa madam, kindly look at this.

Article The Hartung-Knapp-Sidik-Jonkman method for random effects me...