Calculating the amount of underestimation

#1
Coverage rate for a parameter is 91.2%, and the nominal coverage rate is 95%. If the confidence interval is based on asymptotic standard normal, then the amount of coverage 91.2% implies that the standard errors for the parameter is estimated about 15% too small. Because z* value used to make a 91.2% confidence interval is 1.706043 and z* value used to make a 95% confidence interval is 1.96. So, standard error for 91.2% CI is estimated \(\frac{1.706043-1.96}{1.706043}\times 100\%=-14.88\%\), or about 15% too small.

Now my question is if the confidence interval is based on profile likelihood, and I found the coverage rate for the parameter is 91.2%, then how can I calculate how much it is underestimated than nominal 95% ? Will I use standard normal table here also? That is, will there any dissimilarity to calculate how much is it underestimated if the 91.2% CI is profile likelihood interval or 91.2% CI is based on asymptotic standard normal?