Can you show that for all positive integers that are 10 mod 11, none of them have digits that increase at any point? I ran a Python program for numbers up to the millions checking, and I couldn't find any numbers that are 10 mod 11 that also have digits that increase at any point, so I'm inclined to believe it's true.
Sorry, why should the 2nd rightmost digit be $0$? Another idea: if $n$ has non-decreasing digits and at least 2 digits, let $d$ be the leftmost digit. Subtract from $n$ the number whose digits are $dd00\ldots$, with enough $0$s to make it the same length as $n$. The result is congruent to $n$ modulo $11$, still has non-decreasing digits, and has fewer digits than $n$. Keep doing this and we get down to a single digit or $0$, which is obviously not congruent to $10$ modulo $11$.