Not because this brief is so breath-taking (well, maybe), but because your breath paces and punctuates your attention and awareness. Whenever you start to give attention to any awareness of stimulus, you hold your breath! When you breathe again, that is part of a pattern where you are releasing your attention from the one focus of awareness and moving it on to wherever it will alight next.
The claim isn't that conscious control of breathing affects attention (though that might be true), it's that spontaneous changes in breathing shape attention.
Someone without much lung capacity might still have good attention for reading and other low-aerobic effort activities if their metabolism is ramped down enough so that they can get what they're thinking about into their memory.
It wouldn't surprise me if there's a connection between breath and attention, but I suspect it's more complex. I might be able to introspect efficiently enough to find out what triggers spacing out, and it would be worth learning.
On January 4, PJ Eby sent around an email linking an... interesting... website. The claim on the particular webpage he linked was as follows:
This site is part of a sales pitch, so many of the claims are stated in hyperbolic language. I've already noted one factual error: the webpage claims that being underwater triggers the diving reflex, while in fact (or at least, according to Wikipedia) the diving reflex is triggered when one's face is immersed in water colder that 21 °C.
But there is a testable claim here: learn to hold your breath for longer periods of time -- particularly in conditions that elicit the diving reflex -- and you will see increased intelligence. I know that some readers of LW regularly train and test their intelligence, so I offer this as an easily implemented potential method. The possible gains seem to me to outweigh the costs of the training and the low prior probability of the claim.