BUILD_USER_ID not set incase of job triggered with Parameterized Scheduler from upstream job

We noticed that when downstream job triggered by upstream job with Parameterized Scheduler plugin, BUILD_USER_ID is not by the build user vars plugin. 

By looking at this script 

https://github.com/jenkinsci/build-user-vars-plugin/blob/master/src/main/java/org/jenkinsci/plugins/builduser/BuildUser.java

the support is only added for hudson.triggers.TimerTrigger.TimerTriggerCause.

Similarly can we add the support for hudson.triggers.TimerTrigger.TimerTriggerCause also ?