Foulkes' conjecture for the sixth symmetric power

We prove the sixth-symmetric-power case of Foulkes' conjecture. For every integer b ≥ 6 and every finite-dimensional complex vector space V, there is a $\mathrm{GL}(V)$-equivariant injection ${\mathrm{Sym}}^6({\mathrm{Sym}}^{b}V)\hookrightarrow {\mathrm{Sym}}^b({\mathrm{Sym}}^{6}V)$.