From the two matches Murray played this past week at the Queen's Club, where he lost in the second round, we found out little about this unique relationship, beyond the Scot's stamp of approval as an equal opportunity employer.
So what did Murray hope to learn from the first week of their grass-court trial period?
Not much, to be honest.
"I just went on court with her the first time today," Murray said this past Wednesday after his 6-4, 6-4 win over Paul Henri Mathieu -- his first match on grass in nearly a year. "This week there's not going to be any big changes in my game. I also wouldn't expect any before Wimbledon."
It was the last sentence that was meant to resonate with his audience, a room full of British journalists who make a living hanging on every word he utters. The message was clear: Let's all calm down and take a breath. It's going to be a while before we know if Mauresmo was meant for Murray.
A male athlete choosing a woman coach? Who would be so bold? Well, Andy Murray, for one.
On the surface, this seemed quite shocking, even to those who watched Murray grow into a world-class player under his mother Judy's tutelage.
"I have spoken to [ESPN analyst] Darren Cahill quite a lot, and I mentioned to him that I was thinking about possibly a female coach," Murray said. "He thought that Amelie would be a good fit.
"For me, it doesn't feel so different because obviously when I was growing up I had my mom working with me until I was 17 years old," Murray added. "I have always had a strong female influence in my career. I found that with my mom, especially, that she listened extremely well."
Although unusual, it must be said that Murray is not the first top-10 ranked man to have a woman's counsel. Jimmy Connors' mother, Gloria, was never far from his side through most of his career, and Russian Andrei Chesnokov was coached by Tatiana Naumko throughout his 11 years on the tour.
Like Connors and Chesnokov before him, Murray is happy to ignore any ridicule he might receive.
"From other players' points of view, I don't really care whether they think it's a good or bad appointment," he said. "It's whether it works well for me and my team, and hopefully it will be a good move for my career."
In the end, what swayed Murray to Mauresmo were basic elements: likability, communication skills and familiarity with the job details. The two had had a number of conversations, and Murray came away thinking she was the one. As he simply put it: "I liked her."
No one can deny that Mauresmo's resume was solid. She became the first French woman to attain the top world ranking, won Wimbledon and the Australian Open, and, as the current French Fed Cup captain, successfully coached the team back into the 2015 World Group in a very short time.
"It's about the total package that she can offer," Murray said. "In terms of what she achieved on the court, she obviously achieved a lot in her career. I think she's a strong character as well. ... She knows how to win. She was the best in the world."
After taking a few days totally off -- no playing, no gym -- Murray will be out on the practice court this week with Mauresmo and the rest of his team and will be shoring up his game for what he hopes will be a full, two-week stay at Wimbledon. He will also play one exhibition match at Hurlingham, as he did last year.
Among the elements of his game that he's hoping to enhance in an effort to regain comfort on the grass are using his legs more effectively for hitting lower-landing balls, returning first serves more sharply and getting his body more attuned to the proper movement for the surface.
However, what is likely to be even more important than Murray's game being ready for Wimbledon is his being mentally prepared for what lies ahead. From the moment Murray emerged as a potential British champion, his every moment at Wimbledon was scrutinized. When he finally ended Britain's 76-year wait for a homegrown Wimbledon champion last year, it was a thrill of a lifetime for Murray and an entire nation.
The honeymoon period of being the reigning Wimbledon champion, however, is about to come to a close. And in its stead will likely be an even more pronounced eyeing of Murray, as an adoring country waits to see if he's capable of defending his title.
This is an area where Murray believes Mauresmo, who never was able to win her home Grand Slam, has personal experience to impart. She understands what it's like to be watched as if under a microscope.
"I think she was quite open and struggled a bit with the pressure," Murray said. "That can also help. Someone that's been through those experiences themselves, maybe, would have handled things differently."
For Murray fans, the good news is that unlike his trial period with former coach Ivan Lendl, which took place in the privacy of the December offseason, Mauresmo's first few weeks on the job will be on constant display. But it won't be until some time after Murray strikes his final ball at Wimbledon -- no matter when that occurs -- that we'll find out what kind of influence Mauresmo had.