Classification of Jordan multiplicative maps on matrix algebras

Let $M_n(\mathbb{F})$ be the algebra of $n\times n$ matrices over a field $\mathbb{F}$ of characteristic not equal to $2$. If $n\ge 2$, we show that an arbitrary map $\phi :M_n(\mathbb{F})\to M_n(\mathbb{F})$ is Jordan multiplicative, i.e.\ it satisfies the functional equation $$\phi (XY+YX)=\phi (X)\phi (Y)+\phi (Y)\phi (X),\text{for all }X,Y\in M_n(\mathbb{F})$$ if and only if one of the following holds: either $\phi$ is constant, equal to $P\mathrm{/}2$ for some idempotent $P\in M_n(\mathbb{F})$, or there exists an invertible matrix $T\in M_n(\mathbb{F})$ and a ring monomorphism $\omega :\mathbb{F}\to \mathbb{F}$ such that $$\phi (X)=T\omega (X)T^{-1}\text{ or }\phi (X)=T\omega (X{)}^t{T}^{-1},\text{for all }X\in M_n(\mathbb{F}),$$ where $\omega (X)$ denotes the matrix obtained by applying $\omega$ entrywise to $X$. In particular, any Jordan multiplicative map $\phi :M_n(\mathbb{F})\to M_n(\mathbb{F})$ with $\phi (0)=0$ is automatically additive. The analogous characterization fails when $\mathbb{F}$ has characteristic $2$.