Lemma 3.40 on page 46 in Mathas's "Iwahori-Hecke Algebras and the Symmetric Group" states
Suppose that $\lambda$ is an $e$-restricted partition of $n$. Then the ladder tableau $\mathfrak{l}_e^\lambda$ is $e$-restricted.
The proof in the book is confusing, to the point that the author included a new proof in the errata, I've included a screenshot.
I can follow the proof up until the line "By construction, $p$ appears in row... I don't understand how those inequalities/equalities are derived, since the $\nu_i$ seem to be the parts of the partition $\operatorname{Shape}(\mathfrak{t}\downarrow p)$, but they use them to conclude something about $\mathfrak{t}\downarrow m$ instead. My concern is that the parts of the corresponding shapes of these tableaux will be different. If anything, I think my main difficulty is understanding why $$ \nu_1+\cdots+\nu_{r_k+n-p}=\mu_1+\cdots+\mu_{r_k+n-p} $$ implies $\mathfrak{t}\downarrow m=\mathfrak{l}_e^\mu$. After that, I think I can piece the rest together.