Alternating increasing permutations as minimizers

Prove that the limit \(c_{\mathrm{alt}}=\lim_{n\to\infty}\beta(\mathrm{Alt}_n)/n^2\) exists and equals the stated variational expression, establish \(c_-=c_{\mathrm{alt}}\), and show that \(\mathrm{Alt}_k\) minimizes \(\beta\) over all permutations in \(\Pi_k\) for every \(k\).

Background

The quantity $\beta_k^-$ is the minimum expected online embedding time over all permutations of size $k$. Numerical evidence suggests that it is attained by the alternating increasing permutation $\mathrm{Alt}_k=(2,1,4,3,\ldots)$. The conjecture has three progressively stronger parts: existence and an explicit formula for the asymptotic constant, asymptotic optimality, and exact optimality for every finite size.