Groenland, Johnston, Radcliffe and Scott's conjecture for two-element shuffles

For integers $k\leq n$, let $U_k(n)$ be the minimum length of a sequence of independent lazy transpositions whose product maps the elements $1,\dots,k$ uniformly to the $n(n-1)\dots(n-k+1)$ possible ordered $k$-tuples; such a sequence is called a $(k,n)$-shuffle. Groenland, Johnston, Radcliffe and Scott's conjecture. For all $n\geq 2$,

$$U_2(n)=2n-3.$$

The general construction gives $U_k(n)\leq kn-\binom{k+1}{2}$, and the cited work improves this bound for every $k\geq3$. The case $k=2$ remains the conjectured sharp case, and proving it would establish that the displayed upper bound cannot be improved for two-element shuffles.