Putnam 2024 A3

Let $S$ be the set of bijections $$T \colon \{1,2,3\} \times \{1,2,\dots,2024\} \to \{1,2,\dots,6072\}$$ such that $T(1,j) < T(2,j) < T(3,j)$ for all $j \in \{1,2,\dots,2024\}$ and $T(i,j) < T(i,j+1)$ for all $i \in \{1,2,3\}$ and $j \in \{1,2,\dots,2023\}$. Do there exist $a$ and $c$ in $\{1,2,3\}$ and $b$ and $d$ in $\{1,2,\dots,2024\}$ such that the fraction of elements $T$ in $S$ for which $T(a,b) < T(c,d)$ is at least $1/3$ and at most $2/3$?