gotcha - in terms of the analogy, the temperature distribution cannot be any arbitrary distribution with a finite average (i.e. a finite L2 norm) --- including those pathological cases e.g. the temperature is 0 everywhere except at one point x where it takes the value 1 ---- no, the temperature distribution has to have some minimum smoothness structure, meaning that if you adjust the temperature at a single point, you have to also locally adjust the temperature of neighbouring points enough for it to make a difference to the overall global average temperature over the entire space.
for intuition, the animation on the wikipedia "heat equation" page is a good start: https://en.wikipedia.org/wiki/Heat_equation
When you plug initial conditions into the heat equation, over time it evolves to be smoother.
Now the interesting question is, does the conjecture follow from the model itself? Apparently not. But I do neither understand the model nor the proof.