According to signs posted around Volunteer Park, the western loop road will be closing to motor vehicles beginning Friday, June 12th. It will be available for use by pedestrians and cyclists only. Apparently, this will be a full-time deal as part of the Healthy Parks, Healthy You program.
It looks like vehicle-blocking posts will be installed to prohibit traffic at the 12th & Prospect entrance, the loop entrance (just west of the conservatory) and the loop exit (just west of the water tower).
I searched high and low on the Seattle Parks and City of Seattle websites but was unable to find any links to share with further details.
CHS wrote about the closure plan in February — Plan to close Volunteer Park western loop to cars