I was practicing earlier, and I got to thinking... 4-valve compensating horns are technically Bb euphoniums with a valve to switch to F tuba. That F tuba only really has 3 valves. So, the same problem that plagues 3-valve horns comes up in the lower octave- low C and low B are sharp. I figure most players kicked a slide somewhere to remedy this, but I figure there's another, much more complicated way... 5-valve compensating. So I got right to designing and modeling such a system.
It took about 9 hours to model the example (most of that time was figuring out where all the tubing would go). This specimen is 0.578" main bore, for reference. The system is a little bit weird, but hear me out... Valves 1-3 are as usual. Valve 4 is an overbored quart valve (0.586" bore) and is not compensated. Valve 5 is another quart valve, overbored a little more (0.594"), and is compensated. So using the path of logic I used earlier, the open horn is a Bb euph with two quart valves. When the 5th valve goes down, it's a 4-valve F tuba. This way, you can use 4-5 and 2-4-5 for low C and B respectively, and thus they're better in tune.
This arrangement is flexible on the finger work. You can play it as if it were 4+1, or play it as if it were 3+2. Notes have more alternate fingerings, though many aren't useful. Example- low D can be played 2-3-4, 1-2-5, or 3-5. C above the staff can be 3, 1-3, 4, 5, or 4-5. Combination 1-2-3-4-5 could give a very sharp double-pedal G, if you wanted one.
Problems so far: Cost, ease of manufacturing (or lack thereof), and weight. Not to mention, low C and B are hardly ever written in euphonium parts... but if you find yourself in a pinch where there's a tuba part on the stand and a euph in your hands...
even then...
a 4-valve would probably be fine.