Should I use a confidence interval or a prediction interval around the LOESS fitted curve?

A Freakonometrics blog post shows how to use a LOESS regression of the residuals of a logistic model on the predicted values of the logistic model to assess the linearity of the predictors used in the logistic regression model.

If the green interval contains zero, this indicates that the model is correctly specified (or close enough).

My question is, should I use a confidence interval or a prediction interval around the LOESS fitted curve?

I am interested in whether the estimate of $$\text{E}[y \mid x]$$ is correctly specified - so that makes me think confidence interval. However, normally prediction intervals are used for individual observations.

For reference:

Difference between confidence intervals and prediction intervals

How to calculate prediction intervals for loess

• A heuristic I find useful is to ask whether a straight line can fit through the confidence band of a nonparametric regression estimate, including LO(W)ESS: if it can a linear regression straight line estimate is probably as well-fit for the given data, if not, estimation/interpretation of a nonlinear relationship is probably warranted. – Alexis Sep 10 '19 at 16:44
• Possible duplicate of How to calculate prediction intervals for LOESS? – Alexis Sep 10 '19 at 16:45
• Yes, that is the purpose of this plot. Here the straight line is where the mode residuals= 0. It's not a duplicate, I'm not interested in how to make the prediction interval but whether to use the prediction interval or the confidence interval. – Michael Webb Sep 10 '19 at 18:19