A stronger rainbow degree-sequence conjecture for bipartite graphs

Let $F_i$, $i\leq k$, be subgraphs of $K_{n,n}$, and let $A=(a_{ij})$ be the $k\times n$ matrix whose entry $a_{ij}$ is the degree of $v_j$ in $F_i$. Let $r_i(A)$ be the sum of the $i$-th row, so that $r_i(A)=|F_i|$. For sequences, write $\overrightarrow{a}$ for the rearrangement of $a$ in non-decreasing order, and compare sequences componentwise after rearrangement. Stronger rainbow degree-sequence conjecture. If

$$\sum_{i\leq j}\overrightarrow{r}_i>j(j-1)n$$

for every $j\leq k$, then there exists a permutation $\pi:[k]\to[k]$ such that

$$\sum_{i\leq j}\overrightarrow{a}_{i\pi(i)}>j(j-1)$$

for every $j$. This strengthens the preceding theorem by requiring simultaneous inequalities for the rearranged selected columns, rather than only the componentwise lower bound represented by $(1,2,\ldots,k)$.