Narrow it down by checking the rod protrudes when you trip the shutter. If not, a drop of alcohol or Naptha(lighter fluid) should clear it up. If other lenses work, it's likely the back as you say.

I haven't been inside a back for a long time but suspect that would work there too.
A drop, not a squirt. :o) Hold the back vertical so the fluid runs down to the internal linkage.