The finite basis conjecture for polynomial identities of 2 × 2 matrix algebras in characteristic 2

Let $K$ be an infinite field of characteristic $2$, and let $M_2(K)$ denote the associative algebra of $2\times 2$ matrices over $K$.

Finite basis conjecture. The algebra $M_2(K)$ does not have a finite basis of its polynomial identities.

This is a finite basis problem for associative algebras. The preceding examples show that varieties of associative algebras can fail to have the finite basis property, but the conjecture specifically concerns the polynomial identities of the matrix algebra $M_2(K)$ over an infinite field of characteristic $2$.