We prove an upper bound of $1.5321 n \log n$ for the mixing time of the random-to-random insertion shuffle, improving on the best known upper bound of $2 n \log n$. Our proof is based on the analysis of a non-Markovian coupling.
"Improved bounds for the mixing time of the random-to-random shuffle." Electron. Commun. Probab. 22 1 - 7, 2017. https://doi.org/10.1214/17-ECP3955