A genetic test revealed the AKT1 gene mutation.

There are a number of dysplastic features. The cause of the left proptosis turned out sphenoid wing hyperostosis. 

Irregular frontal and temporal bones thickening, meningiomas (1 present, 1 removed) are another one interesting clues that led us to assume Proteus syndrome as a possible cause for it. 

Meningiomas are the most common intracranial tumor in Proteus syndrome and are associated with a characteristic pattern of cranial hyperostosis, which is caused by a somatic activating mutation in AKT1 gene.