Electric guitar intonation flat above 12th fret, no luck with usual methods

Question

My Floyd Rose equipped electric guitar has mostly spot-on intonation across the fretboard minus the high e string above the 12th fret, where it is pretty consistently flat (1-2 red LEDs from the middle on the Snark tuner). The high E saddle is as far forward as it can go, and is in tune on the 12th but flat on the 13th and above. I've tried changing strings, adjusting truss rod for more or less neck relief, and raising and lowering action, with no luck. I don't see any flatness or grooves on those frets visually. I would appreciate any suggestions on what to try next.

Edit: It may be helpful to note that the high e is actually about 7-9 cents sharp from frets 5-8, and 15-25 cents flat from frets 13-24. Every other fret on the high e and all other strings is within 4-5 cents from perfect.

Picture from Comments Section:

Answer 1

Before we start: It is worthwhile to note that all strings must start rom the same height at the peg heads. I had an intonation problem lately and it turned out my B string wasn't starting at the same height as other strings at the peg head.

Given how the string goes Sharp at few places and Flat at other, this case is very different from a Gauge-Change, Saddle problem or a Truss-Adjustment.

Here's Why:

A string's pitch / note being played basically (there are other factors but these are most significant ones) depends on its Length from the fret held and the bridge (called effective length), tension in string and Density of String (Mass of string per Length, not volume). Leave out Tension & Density because they are string specific and you've tried different strings as you say in the question.

Now Length. Normally Intonation goes gradually sharper or gradually flatter. This fact will be obvious if you realize that either the Starting Length (Open note) is more or less than what the Guitar is designed for or the string is at an angle to the fret board different from what it is designed for therefore each fret corresponds to different points on the string than what the guitar is designed for.

Knowing the inner workings, you guitar is going flat at some points and otherwise at others means that this case is not usual, not the Starting Length (Saddle Case), Not the Angle (Truss Rod case). It is most probably the neck itself maybe affected by weather, changed its profile.

Answering your question. Nothing can be said for sure. Best bet would be to Take it to a Luthier, like Tim says in the comments, and it might happen that the luthier will suggest reprofiling the neck. Reprofiling is, Maybe a little given the COVID-19 situations but otherwise, not like very rare and very worrisome a job,. A lot of users here at StackExchange have gotten their guities' necks reprofiled too.