rope is dull and dry  

Briwax ?
Spray paint ?

briwax
so apply briwax on the rope?

it's paper
I don't know that there's much you can do to rejuvenate paper cord. If it were my chair, I'd reweave it. There's a thread here about reweaving paper cord seats. It's way easier than you'd think and also not expensive to do yourself.

Clean and dry
is a good default condition for most materials. Thus, to my way of thinking, good housekeeping need only concern itself with that condition; anything beyond is intended to impress somebody or something: the camera; visitors we wish to impress. . .

That's the same problem with
the Richard Schultz Polyester/Dacron weaved slings for his beautiful Leasure/1966 collection.
Once they get loused up, they cannot be repaired or rewoven.
I have two white slings that have been bleached over the years and the material has weakened and gotten a bit crispy. They cannot be repaired.
Some true wood rattans can be fixed, but rope/wicker made from paper products cannot.

oh, also
It may have been woven with lesser-quality fiber rush and not true Danish paper cord. Fiber rush is just a strip of brown kraft paper twisted into twine---one ply. Danish paper cord is that times three---three ply with each ply twisted. It wears much better.
I have a big spool of old fiber rush that has been sitting around for a few decades and it looks dry and sort of fluffs up when I subject it to very light abrasion.
Neither of these weaving products are made to last as long as the chair frame. Using the chair causes wear and tear. Eventually the cord or rush will break in places.

Soap Flakes
When I got my Wegner chairs the care instructions direct using soap flakes mixed in water as a method of caring for the wood. When I looked up the soap flake manufacturer they also recommend it for papercord seats - see link below:
http://www.soap-flakes.com/recipes.html

yes, soap flakes
Thanks Lucifer. I tend to forget about the many uses of the product.
Another quick QandA here. http://www.msodistributing.com/soapflakes.html
The word soap is misleading.
If the cord is still intact, a few careful applications may help?
We see a fear of maintenance on most furniture over the years.
'Leave it alone or i may screw it up' attitude.
Many products developed in the 70's are criminal. Harsh and toxic.
Silicone based polishes, etc.
Nice to see some good products still available... Bon Ami for stainless,
Borax, etc.
A favorite of mine, BowlingAlleyWax, has gone out of business. : (
http://www.msodistributing.com/soapflakes.html