TAS ubiquity conjecture for oriented paths

An oriented path of length $n$ is a path with $n$ directed edges. Let a fraction be taken over all orientations of the path, and write $o_n(1)$ for a quantity tending to zero as $n$ tends to infinity.

TAS ubiquity conjecture. A $(\frac 1 2 - o_n(1))$ fraction of all oriented paths of length $n$ are TAS.

The preceding results show that only $o_n(1)$ of long oriented paths are TS, while at least $\frac 1 2-o_n(1)$ are neither TS nor TAS. The conjecture asserts that asymptotically this latter proportion is accounted for by TAS orientations.