Kudos to @BojanŠavrič for pointing out the mixed horizontal and vertical reference systems and the need to install the additional Coordinate System Data file.
However, BEWARE there are bugs in the ArcGIS Pro Project Raster tool when using it for vertical transformations. These are confirmed ESRI bugs that were identified when going from NAD 1983 (2011) State Plane Alaska 1 FIPS 5001 (Meters)/NAVD88 (Geoid12b) --> ITRF08 geographic coordinates and ITRF08 ellipsoid heights. The issues go back to at least version 2.4.3 and vary depending on the software version. The issue particular to 2.7 is that the vertical transformation does not occur. The data appears to move correctly in the horizontal but is not being transformed in the vertical. In addition, deleting or retaining the tildes in the transformation strings seems to have no effect (suggesting they are being ignored). As of version 2.8 this bug is still unresolved. [#BUG-000137632].
The bug was only verified for the NAD 1983 (2011)/NAVD88 (Geoid12B) --> ITRF08/ITRF08 case, not when going from EGM96-->NAVD88. I don't know whether the bug persists with different datum transformation combinations, but I would suggest a careful inspection of your output to verify the transformation occurred and the output elevations are what you expect.
For anyone using earlier versions of Pro, a different bug exists in versions 2.4.3 - 2.6.3 [#BUG-000128301, originally submitted 1/31/2020 and never resolved as far as I know]. For the datum case above, going from NAD83(2011)/NAVD88(Geoid12B)-->ITRF08/ITRF08, if you accept the auto-populated, default geographic transformation method (~NAD_1983_2011_To_NAVD88_Alaska_GEOID12B_Height + ~ITRF_2008_To_NAD_1983_2011), your output will be moved horizontally in the wrong direction (the opposite direction it should move) unless you manually delete the tilde in front of the 2nd half of the transformation string (the horizontal portion). Manually deleting this tilde, using an ArcPro version from 2.4.3 - 2.6.3, is the only way I've found to perform a successful horizontal + vertical datum conversion for a raster with ESRI geoprocessing tools.
If you do not delete the 2nd tilde, your output will be moved horizontally in the wrong direction if your input is a geotiff. If your input is a file geodatabase raster and you do not delete the 2nd tilde, the horizontal conversion looks correct, but heights are slightly lower than if you delete the tilde. Complicated, I know...