Nonalgebraicity conjecture for products involving multiple matrix or Grassmann factors

Let $K$ be the ground field, let $E$ denote the Grassmann algebra, and let $R$ be a PI-algebra with

$$T(R)=T(R_1)\cdots T(R_k),$$

where each $R_i$ is one of $K$, $E$, $M_2(K)$, or $E\otimes_K E$, and at least two of the algebras $R_i$ are equal to $M_2(K)$ or $E\otimes_K E$. Nonalgebraicity conjecture. The codimension series $c(R,t)$ is not algebraic. The statement is presented after results completing Regev's conjecture for even $k\geq4$; its resolution status is not specified in the supplied text.