For a partition $\lambda$, let $y_k(\lambda)$ denote the number of Standard Young Tableaux $T$ of shape $\lambda$ satisfying $\mathrm{maj}(T)\equiv 1 \mod k$. (Here $\mathrm{maj}(T)$ is the major index of $T$, the sum of the descents of $T$, where $i$ is a descent of $T$ if $i$ appears in a row above $i+1$ in $T$.)

Then, according to Stanley, Enumerative Combinatorics, Vol. 2, Chapter 7 Exercise 7.88(d), there is no known combinatorial proof of the fact that $y_{n-1}(\lambda) \geq y_n(\lambda)$ for any partition $\lambda\vdash n$, although there is a proof using the representation theory of the symmetric group $S_n$.

I found this by searching for the phrase "no combinatorial proof is known" in EC1 and EC2.