Of course I had to check Helen Azar's book for Olga's diary entry for Wednesday, October 9, 1913. Happily, she mentions Alexei: "Had a Russian and Religion lessons until 11 o'clock. At 11 1/2 we 2 with Papa and Aleksei went to the Shirvansky regiment. There was a moleben and a parade for the 25-th anniversary of the Shirvansky regiment. Took group pictures. Then had breakfast with them at our [place]."
I'd love to see the group pictures she writes about. I suspect the uniform he's wearing in the above photo would match the one he wears in the aforementioned group photo.