The minimal degree conjecture for standard identities of matrix Grassmann algebras

Let $R$ be a commutative unital ring, let $m,n\geq 1$, and let $E^{m}$ be the $m$-generated Grassmann algebra

$$E^{m}=R\langle v_i\mid 1\leq i\leq m\rangle/(v_i^2,v_iv_j+v_jv_i\mid 1\leq i,j\leq m).$$

For $k\geq 1$, the standard polynomial is

$$s_k(x_1,\ldots,x_k)=\sum_{\pi\in\mathfrak{S}_k}\operatorname{sgn}(\pi)x_{\pi(1)}\cdots x_{\pi(k)}.$$

It is a standard identity of degree $k$ for $M_nE^m$ when it vanishes for all $x_1,\ldots,x_k\in M_nE^m$. Minimal degree conjecture. For all $n,m\geq 2$, the minimal degree of a standard identity that is a polynomial identity of $M_nE^m$ is

$$2\left\lfloor\frac{m}{2}\right\rfloor+4n-4.$$

The preceding lower bound shows that degree $2\left\lfloor\frac{m}{2}\right\rfloor+4n-5$ does not suffice, while the degree $4n-2$ upper bound settles the cases $m=2,3$; the conjecture predicts the exact degree for all $m,n\geq 2$.