(Sorry, I assumed the combination gray/black pipe was the rear one, like on my rig.)
No, you eventually have to hook up the rear one, or that gray tank will never get emptied. They're not interconnected.
For long-term camping, I took a (new) blue sewer elbow and bored a hole for a
male hose bulkhead fitting, fed it through from the inside and caulked it all up nicely inside. Now I can hook up my
gray water hose between that fitting (needs
a separate F/F hose connector) and the hose outlet on my
dump pipe cap on the gray-water-only pipe. When I'm using the sewer fitting without the hose connected for a short-term stop, I cap the bulkhead with a hose cap to keep it gas tight.