With about 30% of my annual production in baritones, this subject comes up often. What I find interesting is that subconsciously no one seems to notice the difference in presentation from wound to unwound on regular guitars with the same regularity as they do on baritones. Most often this is on the 6th string that on full scale baritones is double wound and as such has a rounder presentation than the other strings -- its also halfway to a bass and should be beginning to have that presentation.
In your instance, my first question is whether this baritone you are using is one you built or by someone else or production. The EXP23's were designed to go on a 29" scale, so their presentation isn't balanced for other scale lengths. People tend to use the set because its the closest thing to a set of baritone strings, but that doesn't mean its the best choice.
So what is the scale length of your baritone? Is there a difference in output between the 3rd and 2nd string as well? How is the overall string to string balance on the guitar? If the sclae length is anything but a 29" scale then, adjustments are necessary.
I highly recommend against the .022" wound string. The problem here is that it has a very small core wire and will be prone to breakage. You'll also need a new saddle with it because it won't intonate in the same place as your unwound. I'd try going down to a plain .020" or perhaps a .019" to get that string to stop shouting at you. By lowering the tension you'll alleviate the difference between the adjacent strings.
I'd need more information to give you a more complete answer in particular:
Scale length, and how the string to string balance is for all strings, not just the 2nd string. I suspect its not entirely balanced elsewhere as well as the 2nd string.
Finally, there are noticeable differences in presentation between the wound and unwound strings on any guitar. Some of these are native to the instrument, some can be string balance problems, i.e., not the best set of strings for that guitar, and sometimes its that the guitar has a "wolf" note, a note that shouts out more than others and it is this that contributes to the problem. Often that note is problematic not only in the open position, but elsewhere as well, but not always.
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