The identity with constants in a Chevalley group of type $\mathrm F_4$

N. L. Gordeev proved that a generalized group identity holds in Chevalley groups with multiply laced root systems. It was also shown that a stronger identity is valid for the Chevalley groups of types $\mathrm{B}_l$ and $\mathrm{C}_l$. In the present paper, it is proved that this strong identity is fulfilled in Chevalley groups of type $\mathrm{F}_4$ and fails to be true in Chevalley groups of type $\mathrm{G}_2$. The main result of the paper is the last ingredient in the proof of the claim that the lattice of intermediate subgroups between $G(\mathrm{F}_4,R)$ and $G(\mathrm{F}_4,A)$ is standard for an arbitrary pair of rings $R\subseteq A$ with $2$ invertible.