I've heard that Alexander and Sophie, known in the family as "Topy", were rumoured to have been engaged.
However it's hard to believe how credible these rumours might be, considering that the princess, due to post-birth complications, was submitted to health treatments that caused some mental handicap.
She never married and died at the early age of 29 from a cardiac syncope in 1928.
I believe that she was once linked to King Alexander I Karageorgevich of Yugoslavia, and in fact, rumored to have been engaged to him. What was the basis for this rumor? Did those two even meet and form any sort of liaison whatsoever?
Biographical material on him in English is rather scanty, and the sources are not always reliable. But according to Dame Rebecca West (author of BLACK LAMB, GREY FALCON), he also had a romance with Grand Duchess Tatiana Nicolayenva of Russia (second daughter of Czar Nicholas II). She apparently gave him a ring, and the two had an understanding that they would eventually marry.
As it was, the massacre of the Romanovs at the hands of the Bolsheviks (a tragedy which instilled in him a lifelong hatred of communists) prevented him from being united with his great love. Just what exactly her parents thought of him is anybody's guess. We know that his sister, Princess Helen of Serbia, married Prince Joann Constantinovich of Russia (nephew of Queen Olga of Greece). So his status as a member of an upstart royal dynasty should not have been held against him ...
At his end of the deal, he seems woefully ignorant of the risk of marrying such a woman (who carried a 50% chance of being a carrier of the hemophilic gene). Had Alexander (as crown prince) married Tatiana before World War I, she certainly would have been spared the tragic fate that awaited her family.
The same could be said of her older sister, Olga, had she married the then-Crown Prince Carol of Romania. As I understand, the Russian and Romanian royal families had a get-together in 1913, in the hopes of fostering a marriage between those two. Dynastically, it would have been a perfect match. Genetically, not so: Queen Marie of Romania expressed fears that Olga might bring hemophilia into her family.
What hung in the balance, then, was love: if those two were attracted, there was every reason to believe that the risk of her being a hemophilic gene carrier would have been overlooked. As it was, the absence of romantic attraction, along with the 50% likelihood of Olga carrying the gene, made it difficult for the parents to try to arrange a marriage between her and the future King Carol II.
Being a granddaughter of Queen Victoria, Marie would have had a more intimate knowledge of the disease and the genealogical patterns of it appearing in her extended family. The future King Alexander I of Yugoslavia (her future son-in-law), however, would not have been in the same position to observe what she did. Indeed, the Russian imperial couple largely withheld their son's condition from the public.
I know very little about Sophie, other than the fact that she could not have been a carrier of the hemophilic gene. As it was, Alexander was spared the possibility of having it brought into the Karageorgevich family by Tatiana, and married instead a second cousin of hers -- Princess Marie ("Mignon") of Romania (whose older brother was the prince who might have married Tatiana's older sister).