Arguments for/against Kolmogorov's conjecture about the circuit complexity of P
The footnote of my paper that you cite refers to a heuristic "argument" as well, at least, what we think was Kolmogorov's intuition -- the positive resolution of Hilbert's thirteenth problem. 

http://en.wikipedia.org/wiki/Hilbert's_thirteenth_problem

In particular, it was proved by Kolmogorov and Arnold that any continuous function on $n$ variables can be expressed as a composition of $O(n^2)$ "simple" functions: addition of two variables, and continuous functions on one variable. Hence, over the "basis" of one-variable continuous functions and two-variable addition, every continuous function on $n$ variables has "circuit complexity" $O(n^2)$.

It seems Kolmogorov believed there is a discrete analog, where "continuous in $n$ variables" becomes "Boolean in $n$ variables and poly$(n)$-time computable", and where the "basis" given above becomes two-variable Boolean functions.