The asymptotic scarcity conjecture for unique rectification tableaux

Let $I_n$ denote the number of $K$-Knuth equivalence classes of initial tableaux on $[n]$, and let $U_n$ denote the number of unique rectification tableaux (URTs) on $[n]$. Asymptotic scarcity conjecture.

$$\lim_{n\to\infty}\frac{U_n}{I_n}=0.$$

The conjecture is motivated by computed data through $n=7$, where the ratio of URTs to $K$-Knuth classes decreases monotonically. No resolution is given in the supplied text.