The uniform permutation-walk arcsine law

Let a random walk of length $2n+1$ be generated from a uniform random permutation, and let $2k$ be the number of its edges lying above the origin. Write $\alpha_{2n,2k}$ for the corresponding simple-random-walk probability in the discrete arcsine law. Uniform permutation-walk arcsine conjecture. The probability that the walk has $2k$ edges above the origin equals

$$\alpha_{2n,2k}.$$

This conjecture asserts that the relevant edge-count distribution for a uniform permutation-generated random walk coincides with that of a simple random walk; the supplied context reports strong numerical evidence but does not establish the claim.