Brough–Eick exceptional-isomorphism conjecture for Hamiltonian and special algebras

Let $H(4,(m_1,m_2,m_3,1))$ and $S(3,(m_1,m_2,m_3))$ denote the indicated graded Lie algebras of Cartan type in characteristic $2$. Brough–Eick's exceptional-isomorphism conjecture. One has

$$H(4,(m_1,m_2,m_3,1))\cong S(3,(m_1,m_2,m_3)).$$

This extends the two computationally observed isomorphisms listed in the source; its general validity is presented as a conjecture.