If t'were I, I'd have a lash with the solvent - I might even put some on the RIB. I've used general purpose paint thinners and some on here have recommended "gunwash" used for cleaning spraying gear.
All the usual safety stuff: no flames or sparks, good ventilation, gloves. Use lots of spotless rags and do a small section at a time (test a bit underneath first). If it cleans as well as you want, use a protecting product on it before it picks up dirt again. The rubber strip will look good with a touch of trim renovator on it, keep that off the tubes.
As you apply the solvent, you'll feel the fabric getting tacky, don't rub this, but let it dry - it will reharden.