One of the authors here. We are aware of the Winograd scheme, but note that it only works over commutative rings, which means that it's not applicable recursively to larger matrices (and doesn't correspond to a rank 48 factorization of the <4,4,4> matrix multiplication tensor). The MathOverflow answer had a mistake corrected in the comments by Benoit Jacob.

More details: the Winograd scheme computes (x1+ y2 )(x2+ y1 ) + (x3+y4)(x4+y3)-Ai-Bj, and relies on y2y1 (that comes from expanding the first brackets) cancelling with y1y2 in Bj=y1y2 + y3y4. This is fine when working with numbers, but if you want to apply the algorithm recursively to large matrices, on the highest level of recursion you're going to work with 4x4 block matrices (where each block is a big matrix itself), and for matrices Y2Y1 != Y1Y2 (for general matrices).

Here is a website that tracks fastest (recursively applicable) matrix multiplication algorithms for different matrix sizes, and it stands at 49: https://fmm.univ-lille.fr/4x4x4.html

UPD: s/fields/rings/ and fixed equation rendering