Did Ibn Ezra meet the Rambam?

This Shabbos someone pointed out to me a story in one of the parsha sheets. It describes a meeting between Ibn Ezra and the Rambam.

It seems somewhat unlikely that the Rambam had a nurse, and further, that he had a female nurse.

But aside from that, I wondered whether this story, which has many signs of being an all-too-perfect legend, could have happened. That is, what is the overlap of their lifetimes? Ibn Ezra traveled to many places, but was he in Egypt (where the story is presumably set) at the same time as the Rambam?

I found a version of this story a bit more fleshed out here, but without sources cited.

Rav Avraham’s journeys brought him as far as Egypt, where he met the Rambam.
The Rambam served as the court doctor, and he also built a special hospital in Cairo for the country’s Jewish residents. He personally attended to each of the patients and personally subsidized the costs of their treatments. 

Interested in observing the Rambam’s behavior firsthand, Rav Avraham one day entered the hospital and waited to be accepted. When one of the doctors asked him what was wrong, he complained about various aches and pains so that he would be permitted to remain in the hospital for a while. 

As Rav Avraham lay in his hospital bed, he watched the Rambam go from patient to patient, diagnosing their illnesses and attaching notes to their beds, including Rav
Avraham’s, with instructions for the staff. 

Rav Avraham, who really wasn’t ill, wondered why the Rambam had reached a diagnosis and had prescribed medications. After he read the note, he was truly astounded by the Rambam’s brilliance. It read, "Illness: poverty. Cure:
four hundred dinars."

I see the story here as well.

Just looking at Wikipedia, this is what I discovered.

Ibn Ezra's life and travels:

He was born at Tudela,^{[citation needed]} (in the present-day province of Navarre) when the town was under the Muslim rule of the emirs of Zaragoza. Later he lived in Córdoba. In Granada, it is said, he met his future friend (and perhaps his father-in-law) Yehuda Halevi. He left Spain before 1140 to escape persecution of the Jews by the new fanatical regime of the Almohads. He led a life of restless wandering, which took him to North Africa, Egypt (in 1109, maybe in the company of Yehuda Halevi), the Land of Israel, Italy (Rome in 1140-1143, Lucca, Mantua, Verona), Southern France (Rodez, Narbonne, Béziers), Northern France (Dreux), England (London, and Oxford in 1158), and back again to Narbonne in 1161, until his death on January 23 or 28, 1167, the exact location unknown: maybe at Calahorra at the border of Navarre and Aragon, or maybe in Rome or in the Holy Land. There is a legend that he died in England from a fever and a sickness which came upon him after an encounter with a pack of wild black dogs. This legend is attached to the belief that he denied the existence of demons.^[4]

So he was in Egypt from 1109, was in Israel, and was in Rome from 1140-1143. And he died in 1167.

And the Rambam:

He was born in Córdoba (present-day Spain), Almoravid Empire on Passover Eve, 1135, and died in Egypt on December 12, 1204.

and

For the next ten years, Maimonides moved about in southern Spain, eventually settling in Fes in Morocco. There he studied at the University of Al-Karaouine. During this time, he composed his acclaimed commentary on the Mishnah in the years 1166–1168.^[18]

Following this sojourn in Morocco, together with two sons,^[19] he sojourned in the Holy Land, before settling in Fostat, Egypt around 1168. While in Cairo, he studied in Yeshiva attached to a small synagogue (which now bears his name).^[20] In the Holy Land, he prayed at the Temple Mount. He wrote that this day of visiting the Temple Mount was a day of holiness for him and his descendants.

So the Rambam settled in Egypt in 1168, which was one year after the death of Ibn Ezra. Maybe one could suggest a different place for their meet, such as Eretz Yisrael, or Morocco?

Also, when Ibn Ezra actually visited Egypt, it was 1109-sometime before 1140, before Rambam was born until when the Rambam was only five years old, and presumably not a predecessor of Doogie Howser, MD.

Unless I am missing something, which is certainly possible, this indicates to me that someone thought to make up a good story in which the two great rabbis met, and invented it without regard for the actual timelines of their lives.