Tuning Mystery

4 string dulcimer, 27" vsl, melody strings 1 & 2 .012; string 3 .014, string 4 .020.  

Problem: when tuned ddaD all strings are in tune when open. Strings 1,2 & 3 remain in tune all the way up the scale. String 4, bass string, is consistently sharp all the way up the scale, when starting with E (first fretted note).  If I retune #4 to be in tune starting with E it remains in tune all the way up the scale, but D is flat. 

All frets are parallel with one another and when I measure the vsl on either side of the fretboard it is exactly 27" so the nut and bridge are parallel. 

Any suggestions what is wrong? Any comments are appreciated.