The density-one conjecture for numbers never reaching a v-palindromic number

Let $r(n)$ denote the reversal of the decimal digits of $n$, and define

$$S=\{n\in\mathbb{N}:10\nmid n,\ n<r(n)\}.$$

Let $c(n)$ be the least number of repetitions of the decimal digits of $n$ needed to reach a $v$-palindromic number, with $c(n)=\infty$ when no such repetition reaches one, and define

$$T=\{n\in S:c(n)=\infty\}.$$

Density-one conjecture. The asymptotic density of $T$ in $S$ is $1$. The paper proves that an explicitly described infinite family has $c(n)=\infty$, and asks for a simple way to determine whether $c(n)$ is finite. The stronger assertion that almost every element of $S$ belongs to $T$ remains open.