The neural correlates supporting our perceptual experience of the world remain largely unknown. Recent studies have shown how stimulus detection and related confidence involve evidence accumulation (EA) processes similar to those involved in perceptual decision-making. Here, we propose that independently from any tasks, percepts are not static but fade in and out of consciousness according to the dynamics of a leaky evidence accumulation process (LEAP), and that confidence corresponds to the maximal evidence accumulated by this process. We discuss the implications and limitations of our proposal, assess how it may qualify as a neural correlate of consciousness, and illustrate how it brings us closer to a mechanistic understanding of phenomenal aspects of perceptual experience like intensity and duration, beyond mere detection.