Latin squares puzzle

A Latin square is an n x n array containing the numbers 1 to n exactly once in each row and exactly once in each column.

For example, L is a Latin square:

123 231 312

L

A critical set C in a Latin square is a subset of the Latin square which is a subset of exactly one Latin square, and has the property that every proper subset of C is a subset of more than one Latin square.

For example

1.. ... ...

is a subset of both

123 132 231 and 213 312 321

but

1.. .3. ...

is a critical set because it is a subset only of the Latin square L above.

Now we can have four disjoint critical sets in a Latin square such that the union of the four sets is that Latin square:

1. .2 .. .. .. .. 2. .1

And we can have three disjoint critical sets in a Latin square such that the union of the three sets is that Latin square:

.2.. ..3. 1..4 2.4. .3.1 .... ..1. ...2 34.. .1.. ..2. 4..3

The puzzle is to find two disjoint critical sets in a Latin square such that the union is that Latin square.

Has anyone attempted this problem? I have a solution (in a forthcoming paper), but I find it strange that I got no replies.

I thought that people would try even harder to solve something no-one else could solve. You know, to prove superiority. Especially on rec.puzzles.