Accessing file geodatabase (.gdb) rasters with GDAL?

Reading and writing file geodatabase rasters is not supported by the File Geodatabase API as of July 2015. According to this 2010 Esri blog File Geodatabase API details:

Rasters (Raster Dataset, Raster Catalog, Mosaic Datasets and Raster Attributes) are not supported with the initial release of the File Geodatabase API.

Additionally, according to an Esri Staff member on a Geonet discussion (June 2015):

Currently we do not have any plans on supported access to raster data via the File Geodatabase API. If you feel this is needed, please add it to the Esri Ideas site and it will be considered for a…