The hit-and-run version of top-to-random

We study an example of a {\em hit-and-run} random walk on the symmetric group $\mathbf S_n$. Our starting point is the well understood {\em top-to-random} shuffle. In the hit-and-run version, at each {\em single step}, after picking the point of insertion, $j$, uniformly at random in $\{1,\dots,n\}$, the top card is inserted in the $j$-th position $k$ times in a row where $k$ is uniform in $\{0,1,\dots,j-1\}$. The question is, does this accelerate mixing significantly or not? We show that, in $L^2$ and sup-norm, this accelerates mixing at most by a constant factor (independent of $n$). Analyzing this problem in total variation is an interesting open question.
Comment: 18 pages, 2 figures