Does $K_{70,70}$ decompose into subgraphs isomorphic to $K_{1,1}$ through $K_{24,24}$?

The complete bipartite graph $K_{n,n}$ has $n^2$ edges. There's a curious number quirk that $$70^2=4900=1^2+2^2+\cdots+24^2.$$ This motivates the question:

Question: Does $K_{70,70}$ decompose into subgraphs isomorphic to $K_{1,1}$ through $K_{24,24}$?

I just ask out of simple curiosity since the number of edges checks out.

It's equivalent to asking for a $70 \times 70$ matrix which contains symbols $1,\ldots,24$ where symbol $i$ belongs to an $i \times i$ all-$i$ matrix.

One sanity check: If it were possible, for each vertex in one part, we could write down the values of $k$ for which it intersects the $K_{k,k}$ used to decompose $K_{70,70}$. This list would contain $1$ through $24$, with the number $i$ occurring exactly $i$ times, and no repeats in the rows. I found an example of such a list:

[1, 22, 23, 24], [2, 21, 23, 24], [2, 21, 23, 24], [3, 10, 16, 18, 23], [3, 14, 16, 17, 20], [3, 20, 23, 24], [4, 14, 15, 16, 21], [4, 15, 16, 17, 18], [4, 21, 22, 23], [4, 21, 22, 23], [5, 13, 15, 16, 21], [5, 19, 22, 24], [5, 20, 21, 24], [5, 20, 22, 23], [5, 20, 22, 23], [6, 13, 14, 16, 21], [6, 13, 14, 18, 19], [6, 17, 23, 24], [6, 19, 22, 23], [6, 19, 22, 23], [6, 20, 21, 23], [7, 12, 15, 16, 20], [7, 13, 14, 15, 21], [7, 18, 21, 24], [7, 18, 21, 24], [7, 18, 22, 23], [7, 18, 22, 23], [7, 20, 21, 22], [8, 10, 13, 17, 22], [8, 11, 12, 15, 24], [8, 11, 13, 17, 21], [8, 13, 15, 16, 18], [8, 17, 22, 23], [8, 19, 20, 23], [8, 19, 20, 23], [8, 19, 20, 23], [9, 10, 12, 15, 24], [9, 11, 12, 17, 21], [9, 12, 13, 17, 19], [9, 17, 20, 24], [9, 17, 20, 24], [9, 18, 19, 24], [9, 18, 19, 24], [9, 18, 21, 22], [9, 19, 20, 22], [10, 12, 14, 16, 18], [10, 15, 21, 24], [10, 15, 21, 24], [10, 17, 19, 24], [10, 17, 20, 23], [10, 18, 20, 22], [10, 18, 20, 22], [11, 12, 13, 15, 19], [11, 12, 14, 16, 17], [11, 12, 14, 16, 17], [11, 12, 23, 24], [11, 13, 22, 24], [11, 16, 19, 24], [11, 16, 19, 24], [11, 16, 21, 22], [12, 14, 20, 24], [12, 14, 21, 23], [13, 14, 19, 24], [13, 15, 20, 22], [13, 17, 18, 22], [14, 15, 18, 23], [14, 15, 19, 22], [14, 15, 20, 21], [16, 17, 18, 19], [16, 17, 18, 19]

This means arguments involving degrees alone cannot exclude this possibility.

In the matrix equivalence, this means that we can specify the columns the all-$i$ submatrices intersect in a non-clashing way (but it doesn't simultaneously give the rows).

Such decomposition does exist.

It was constructed in the answer to this MO question.

Note, that the complete answer there is formed by two answers by different users: the answer by @RobPratt gives us the explicit construction of such decomposition and the answer by @dvitek provides us with the theoretical base for this construction.