doctor who - Was the White Guardian involved in the Key to TIme at all?

In the "Key to Time" epic, covering the whole of season sixteen of Doctor Who, the Doctor is sent on a mission to retrieve the six pieces of the Key, ostensibly by the White Guardian. However, he is not the only person looking for the pieces.

The third story, "The Stones of Blood," begins with a warning voice: "Beware of the Black Guardian," which is the first mention of the Guardians since Romana's arrival in "The Ribos Operation." The Doctor then visits contemporary Earth, where the third segment has been held by the alien criminal Cessair** of Diplos, having brought it there thousands of years earlier. And Cessair knows both how to use the segment's shape-changing powers to alter her appearance, and that the Doctor is there looking for the segment. The natural conclusion is that she is an agent of the Black Guardian, who is seeking the key.

In the last story, "The Armageddon Factor," the main villain, the Shadow, makes no attempt to hide the fact that he is a servant of the Black Guardian, and that he has been waiting for the Doctor to arrive with the first five segments of the Key. ("I am the Shadow. Your adversary, shall we say. It is not important. You come in quest of a key.") At the end, with Shadow, in death, calls out to his master for help, but the Black Guardian disguises himself as the White and goes to claim the complete Key to Time from the Doctor and Romana.

My inference, from the first time I saw the stories, was that the whole storyline was entirely the work of the Black Guardian, that the figure who gave the Doctor his assignment in the first story was not the White Guardian, but again the Black in disguise. So the Doctor, Cessair, and the Shadow were all working for the same power, knowingly of not. This explains, for example, the curious fact that Cessair of Diplos had captured one of the segments and brought it to the Doctor's favorite planet; it was there waiting for him to swing by and claim it.

However, in a number of online discussions of the season-long plot, I have seen the suggestion that the White Guardian really did put the Doctor on the job, and that he completed his adjustments to correct the coming chaos in spacetime while the Key was complete, before it was nearly stolen by the Black Guardian. My ultimate question is whether there is there any evidence for this idea in the stories? Or is this just a product of people misunderstanding a somewhat subtle plot?

** Only today, as I was researching this question, did I learn that "Cessair," like several the villainess's other identities, is the name of a Celtic heroine/goddess.