Armstrong's conjecture on the longest k-alternating subsequence

Let $\pi$ be uniformly distributed in ${\mathcal S}_n$, and let $L_{n,k}=L_{n,k}(\pi)$ denote the length of its longest $k$-alternating subsequence, where a subsequence is $k$-alternating if its successive entries alternate in direction and each successive jump has absolute value at least $k$. Armstrong's conjecture. For all $n\geq 2$ and $k\in\{1,\ldots,n-1\}$,

$$\mathbb{E}L_{n,k}=\frac{4(n-k)+5}{6}.$$

Armstrong made this conjecture and verified it by exact computation for certain small values of $n$ and $k$. The case $k=1$ recovers the known mean of the longest alternating subsequence; the general assertion remains open in the source.