What are "unspecified" differentials in a differential characteristic?
In these papers they are trying to search for a strong boomerang trail and are semi-exhaustively searching over families of trails where each family has a higher chance of holding than a trail chosen uniformly at random from all possible trails. In these families the unspecified differentials are what we (semi-)exhaust over and the specified parts of the differential are fixed for the family and improve the probability of a typical member of the family above uniform random.
In your example we are looking at two families of differential trails (one upper and one lower) where the fixed differentials for the upper family are lime and white and the fixed differentials for the lower family are pink and white. The fixed parts of the differentials were likely chosen because they have a very high probability of holding and the maximum probability of differentials on the unspecified parts was not as strong. Greedily choosing likely values of the grey differentials gives a large collection of high probability trails for both the upper and lower families. These can then be efficiently pairwise tested using a meet-in-the-middle approach to calculate the probability of the trails combining to make a strong boomerang trail using e.g. a Boomerang Connectivity Table. Trying enough upper and lower trails will hopefully result in a high probability upper trail that combines well with high probability lower trail and a very strong overall boomerang distinguisher.
Note that if more than one strong trail is found, the probabilities reinforce as the same statistical test will detect any trail that has the same start and end points.