Quadratic upper-bound conjecture for random card shuffling

Let $D$ be a deck with $n$ cards of each type, and let $\lambda_D$ denote the expected number of steps to absorption for the random card-shuffling process started from $D$. As $n\to\infty$, quadratic upper-bound conjecture. There is an absolute constant $C\leq 2$ such that

$$\max\{\lambda_D\colon D\text{ any deck}\}\leq Cn^2+o(n^2).$$

The conjecture is motivated by experimental data for decks with $n=26$ and asserts that the worst-case expected absorption time is asymptotically at most quadratic in $n$, with leading constant no greater than $2$; its status is not established in the supplied source.