  Quantum nonlocality based on state discrimination describes the global
property of the set of orthogonal states and has a wide range of applications
in quantum cryptographic protocols. Strongest nonlocality is the strongest form
of quantum nonlocality recently presented in multipartite quantum systems: a
set of orthogonal multipartite quantum states is strongest nonlocal if the only
orthogonality-preserving local measurements on the subsystems in every
bipartition are trivial. In this work, we found a construction of strongest
nonlocal sets in $\mathbb{C}^{d_{1}}\otimes \mathbb{C}^{d_{2}}\otimes
\mathbb{C}^{d_{3}}$ $(2\leq d_{1}\leq d_{2}\leq d_{3})$ of size $d_2d_3+1$
without stopper states. Then we obtain the strongest nonlocal sets in
four-partite systems with $d^3+1$ orthogonal states in $\mathbb{C}^d\otimes
\mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}$ $(d\geq2)$ and
$d_{2}d_{3}d_{4}+1$ orthogonal states in $\mathbb{C}^{d_{1}}\otimes
\mathbb{C}^{d_{2}}\otimes \mathbb{C}^{d_{3}}\otimes \mathbb{C}^{d_{4}}$ $(2\leq
d_{1}\leq d_{2}\leq d_{3}\leq d_{4})$. Surprisingly, the number of the elements
in all above constructions perfectly reaches the recent conjectured lower bound
and reduces the size of the strongest nonlocal set in $\mathbb{C}^{d}\otimes
\mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}$ of
[\href{https://doi.org/10.1103/PhysRevA.108.062407}{Phys. Rev. A \textbf{108},
062407 (2023)}] by $d-2$. In particular, the general optimal construction of
the strongest nonlocal set in four-partite system is completely solved for the
first time, which further highlights the theory of quantum nonlocality from the
perspective of state discrimination.
