So now that both Broly, and Kale, who has a transformation which looks exactly like Broly's, have been quoting saying both the same "My Ki is building... Overflowing..." and that Kyabe tells Kale "stop gathering ki!", this topic comes to talk again. Is Broly and now Kale legendary super saiyan transformation limitless in power? This has been talked before with Broly's movies, but the last episode of Dragon Ball Super brings new information to the topic. What about the other canon sources like the Daizenshuu, does it say something about it?
It seems entirely feasible for them to have an unlimited supply of ki. Perhaps while in that transformation, their bodies rapidly generate more and more ki until it just overflows. That however isn't the same thing as having limitless power. Just because you have infinite MP, doesn't mean that you can deal a lot of damage.
To answer your question, I'm not aware of the Daizenshuu mentioning it (nor any official source), but it is clear that the legendary super saiyan transformation is different than the normal transformation. Unlimited ki has appeared in Dragon Ball before too, Androids 17 & 18 were stated to have such an ability. Thus, given the precedence and the dialogue we've seen in regards to the form, I can say with absolute certainty that the answer is "probably".