Running Titan Rune Alpha or Beta is the same as running a heroic.

After running DTK in Titan Rune Beta and being saved, Titan Rune Alpha and Lick King Heroic Dungeons should display saved dungeon information for all 3 categories. Currently it only displays on the dungeon that was run, in this case Titan Rune Beta.

The screenshots below shows Titan Rune Beta saved info but does not show the completed (saved) dungeon information on Alpha and Heroic:

Titan Rune Beta will be in RED color and has the tooltip. The other categories Titan Rune Alpha and Heroic Dungeon will be shown in orange to indicate a LOCKED state with no tooltip info.