n = 1, here is my marathon progression, I follow high fat diet and high volume (lsd)

2008 first ever marathon, 3:54
2011 (june): 3:06
2012 (feb): 3:01
2012 (june): 2:59
2014 (april): 2:54

Have we laid that to bed for good? nailed the coffin shut? Gave it the old heave ho once and for all?

Haven't seen or heard anyone bring it up in ages. (thankfully)

Anyone still holding onto that belief or any coaches still preaching that?

This means improving your ability to do work with less energy. There are many ways to improve your economy, from more effective training to improvements to your aero position on the bike to reducing your rolling resistance with a better set of tires. In my opinion the best way to improve your economy on the bike is to train in the morning before you eat, while in a “fasted” state. Get up early, make sure you have water in your water bottles, and do a workout before you eat, forcing your body to burn its fat stores. Take it at fairly low intensity for the first couple hours at your endurance pace. After two hours, start eating; just make sure you have plenty of protein along with complex carbohydrates for your on-the-bike breakfast. Clearly, doing longer rides enhances this training more than just an hour’s ride before work, but if you’ve got only an hour to ride, then do your hour-long workout and resist the temptation to eat for another hour and half if possible; this will help stimulate your body’s reliance on fat as a fuel source.
If you can make these simple additions to your training, you’ll be able to improve your aerobic engine, the biggest benefit of which is being able to burn more fat as fuel at a higher rate of intensity. Your aerobic metabolism is the bedrock of your cycling fitness, whether you’re an Ironman or a sprint triathlete. When we talk about triathletes with “big engines,” what we’re really saying is they have extremely well-developed aerobic metabolic systems. Think of it this way: the stronger your aerobic energy system (especially your fat metabolism), the less your body will rely on carbohydrate use while riding at a set workload (e.g. 200 watts) early on in a race. This allows you to ride longer at that 200 watts before running out of glycogen or have more carbohydrates available for the hard attacks above 200 watts at the end of a race. In day-to-day training, a stronger aerobic metabolism thus also becomes the critical foundation for improving your anaerobic capacity by permitting you to do more or harder intervals. Therefore, if you can do just one thing to improve in cycling, you’ll want to train to improve your body’s aerobic system and its efficiency at using fat, carbohydrates, and protein as fuel, because the more fat you can burn while cycling at a moderate intensity, the more you can save precious glycogen for more intense running at the end of your event.