Some concerns on the border rank of Kronecker products of the Coppersmith-Winograd tensor

This note provides a detailed proof of Conner–Gesmundo–Landsberg–Ventura’s result that the border rank of the Kronecker square of the little Coppersmith–Winograd tensor is $(q+2{)}^2$.We also indicate how the same ideas seem to extend to the case of the Kronecker cube, pointing toward the conjectural value $(q+2{)}^m$ for $m\ge 4$, although a full proof is left for future work.