Embedded servlet customizer unavailable to management webserver

I have a custom WebServerFactoryCustomizer but its not available for management webserver. What's the best way to make the customization available to management webserver?

here is the customizer (to disable TRACE for Undertow)

public class UndertowCustomizer implements
  WebServerFactoryCustomizer<ConfigurableUndertowWebServerFactory> {

@Override
public void customize(final ConfigurableUndertowWebServerFactory undertowWebServerFactory) {
  undertowWebServerFactory.addDeploymentInfoCustomizers(deploymentInfo ->
      deploymentInfo.addInitialHandlerChainWrapper(handler ->
          new DisallowedMethodsHandler(handler, HttpString.tryFromString(HttpMethod.TRACE.name())))
  );
 }
}

A reproducible test case is at https://github.com/ranarula/WebServerCustomizer

Filing the ticket based on the discussion with @snicoll & @bclozel on Gitter

Start by reproducing the issue with the linked WebServerCustomizer example and its UndertowCustomizer. Trace how WebServerFactoryCustomizer is applied to the main and management webservers, then verify that the documented or implemented behavior makes the customizer available to the management webserver as well.