About six years ago I bought a Heimatklang F alphorn on eBay. The company would put one horn a month on eBay with a starting bid of $100 and just see what it goes for (I guess). I managed to snipe one at about $700. So for the price and all things considered, I'm very happy.
The downside is that, while the horn is overall a bit sharper than the 440 tuning it was advertised to have (it's probably closer to 444, but whatever, since I intend to use it in solo settings), the high concert F (2nd ledger above bass staff; octave above middle C in transposed pitch) is 40-50 cents flat. I *can* lip it up, but only after articulating the note, because if I try to hit the note in tune, it's cracktown.
The issue with alphorns is they go with size by inside diameter of the rim; there's not much discussion over cup shape or size. The mouthpieces aren't marked either, so basically they guess mine is a Lechgold mouthpiece, and I measured the inner diameter of the rim to be 25mm. After a ton of trying to explain that 50 cents is way too flat to be acceptable, a trombone professor who doubles on alphorn suggested I contact this Swiss mouthpiece maker, and get a smaller rim diameter, and a "V cup" (I'm guessing same as funnel). My current piece is 25mm with a bowl shape, and he suggested I go with 23-24mm. He gave me a bunch of names of mouthpiece makers and I went to Google...which was very difficult since their names weren't the domain name, and they're all in German, so Googling was a lot of work. His reasoning is that the alphorn is closer in range to an alto trombone or French horn, so I should be playing something smaller with a different shaped bowl. This seems like sound advice.
The mouthpiece maker he suggested actually does talk about cup shapes, but not any actual measurements of the cup. I sent this guy a message with the video below. He says he doesn't believe a mouthpiece will help, and that it's probably just a bad horn.
This is where I'm confused. Since the alphorn is just a straight conical bore, and I don't see any actual imperfections in the instrument, I can't imagine what in the actual instrument design could cause a note that's 3 octaves above the fundamental tone to be so flat. The 3rd or 5th being so out of tune *may* be believable, but this doesn't seem to gel with what I know about acoustics.
So, I'm asking brass players who seem to speculate more about the mechanics of a labrophone and what I might be able to do to fix it short of buying a new horn or spending just as much on an alternative solution. If this were a euphonium, I'd guess there was a leak somewhere, but as far as I know, there's nowhere for air to escape.
I think I ruffled a few feathers in that discussion, because I mentioned I'm not really looking to master the instrument per se, but would like to pick up some Oktoberfest gigs with it, since hell, I do own the instrument. I just don't want to sound like an amateur in performances.
I think even Chewie hates that note.

Thoughts?