Is this hopeless?  

I dont see why not.
are you sure it is mold? I don't see why it could not be scrubbed down.

It's not hopeless but I'm afr...
It's not hopeless but I'm afraid you won't succeed in getting back the original condition, seems the smut has deeply penetrated the veneer.
First try to treat it by using a damped scrub sponge with fine wire wool, if the result is not satisfying continue with fine sandpaper (180-200), be careful and don't remove too much of the veneer, then you will see how deep it's penetrated.
Afterwards a new varnish and it will look better.
But treat the whole seat and not only the stained spot.

Try bleach.
The regular household/laundry variety mixed 50/50 with water on a cotton swab. Gently rub a small area maybe a couple of square inches, rinse well with water, and blot dry. If the stains are indeed mildew, you should see almost immediate results.

I'm glad to hear that it's no...
I'm glad to hear that it's not as hopeless as I initially thought. Would bleach affect the finish though? Should I oil the panels afterwards?
This maybe a separate question, but I would like to know how others take care of the wood panels on their plywood furniture. On the rosewood of the 670/671, people oil the panels periodically, but I don't know if people do that to their other plywood furniture. Thanks.

.
pubpat isn't often right, and he's wrong again.
Do NOT sand.
If the dirt or mold is that far ingrained then any sanding to reach it will just go through to the next layer of veneer equalling one ruined chair. First try scrubbing (toothbrush, nail brush that kind of thing) with white spirit (naphtha) to avoid raising the grain and allow to dry completely. If no joy you may need to remove the existing finish, whatever it is, and then repeat the scrubbing with naphtha. Afterwards bleach if necessary but remember this will raise the grain and you will also need to match the back rest as the seat will be paler. I would be more inclined to use oxalic acid than household bleach. Allow to dry and refinish. Before applying a finish it is OK to very lightly sand (400 grit max) but this will be simply to smooth any raised grain.

Absolutely!
sanding would be chair suicide
Whatever that is..dirt, mould, is ingrained - the clue is in the word.
You'd be sanding until you reached somewhere you really wouldn't want to be.
Take Paulanna's advice, please.

Thanks everyone for your...
Thanks everyone for your input, including pubpat. Ok, here's what I'll do: I'll go get some mineral spirits and use a toothbrush to see if it can remove the mold. If not, I'll use a fine steel wool in place of the toothbrush, or is that too rough? If that doesn't work, how do I remove the existing finish to proceed further?
I look forward to this fun challenge!

What about putting it out in the
sun? UV rays will kill mold, right? I mean just to kill the stuff, not remove the staining, though sun will lighten some of that.

More detail about the damage
Seller said the damage came from someone leaving a wet towel on the seat for an extended period of time. I believe it's mostly the stain that's remained. So spanky, I don't think the UV solution would be applicable.

as suggested above
water marks often come right out with oxalic acid, barkeeps friend (suggested by someone here at some point) is a great way to suspend/apply the stuff. I have had crazy good luck w/ it

Where can I find oxalic...
Where can I find oxalic acid? I just bought from mineral spirits from Home Depot so I want to get all the stuff necessary before I move forward. Thanks.

I BELIEVE
Barkeep's Friend has a weak solution of it. IT's also available at HD, it's cheap, and it's already a paste, good for rubbing and coating. I slather it on, let it dry, wipe it off.

sherwin williams paint
sells oxalic acid as "wood bleach" where I live.

This is a before/after of a s...
This is a before/after of a stain removed with the oxalic acid in Bar Keepers Friend cleanser.