There are two methods to calculate a confidence interval (CI) around a measure of effect size: you can either construct a CI around the noncentrality parameter of the corresponding noncentral distribution, or bootstrap an empirical distribution of the measure.
I wonder which one to choose for Cohen's $w$. There is some reasearch on which method to choose for Cohen's $d$, but (as far as I can see) none on Cohen's $w$.
Cohen's $d$ relies on some assumptions as to the data, e. g. normality, and the bootsrap method seems to be more robust to violations of these assumptions than the noncentrality method (e. g. The sensitivity of three methods to nonnormality and unequal variances in interval estimation of effect sizes). Hence, for Cohen's $d$ the bootstrap method seems the better choice.
Waht are the crucial assumptions for Cohen's $w$? To my mind, there is only one, namely that under $H_1$ the test statistic of the chi-squared test must follow a noncentral chi-squared distribution. And this is true if under $H_0$ the test statistic is (approximately) chi-squared distributed.
Thus, if the assumptions of the chi-squared test are given, both methods should lead to the same result. Am I right?