Before I post my question, let me set up some notation.
Notation. For $k\geq 1$, let $\lambda \vdash k$ be a partition of $[k]$. Let $C(k,m)$ be the set of all partitions $\lambda \vdash k$ of size $m$. A semistandard Young tableau with weight $\beta\in {C}(k,m)$ is a Young tableau with numbers from a word with composition $\beta$ (so possibly repeated) each in one cell such that the numbers are weakly increasing to the right and strictly increasing downwards. Let $f_\lambda$ denote the number of Young tableaux of shape $\lambda\vdash k$ over the alphabet $[k]$, and let $K_{\lambda,\beta}$ be the Kostka number, i.e., the number of semistandard Young tableaux of shape $\lambda$ and weight $\beta$.
Given a composition $\beta\in \mathcal{C}(k,m)$, let $S_\beta := \{(i_1,\dots,i_k)\in[m]^k \text{ corresponds to } \beta$}. Intuitively $S_\beta$ is the following: given a word $a\in[m]^k$, there is an associated composition $\beta\in C(k,m)$, where $\beta_i$ is the frequency of the letter $i$ in $a$) and $S_{\beta}$ is the set of all possible words $a \in [m]^k$ for which the associated composition is $\beta$. Notice that $|S_\beta| = k!/\beta!$ where $\beta! := \prod_{i=1}^{m}\beta_i!$.
Question Then, is the following true?
$$ \text{ For every } \beta \in C(k,m),\quad \sum_{\lambda\vdash k} \frac{K_{\lambda,\beta}f_\lambda }{|S_\beta|} \log\frac{f_\lambda^2 }{|S_\beta|} = O(\log{|C(k,m)|}) $$
Firstly, it is unclear if the above inequality is true (by small numerical simulations it does seem true).
If the general case is hard, we are concerned with the scenario where $m=O(k)$, say $m=100k$ for example, then we'd like the upper bound RHS above to be $O(k)$ instead of the trivial $O(k\log (m+k))$. If it were true, the proof would have an interesting consequence in matrix theory (which I'd be happy to share if it could be proven).
