Numerous superdeformed (SD) bands have been identified and studied in various domains of the nuclear chart. An example of recent progress in this research area is the discovery of the SD states in a very light mass region ( A ∼ 40) such as $^{36}$Ar [1], $^{40}$Ca [2], and $^{44}$Ti [3]. A striking feature of these experimental works is that the SD band members were assigned from band heads up to high spin states. Discrete γ transitions linked those SD bands to spherical or normal-deformed states, which established the excitation energies and the spin-parities of the SD bands. Theoretical microscopic studies of deformed states in the A ∼ 40 region have been per- formed based on the shell model [4, 5], mean-field models [6–8], and the antisymmetrized molecular dynamics (AMD) [9–14]. Calculations with no assumption for the axial symme- try suggest the systematic triaxial deformation of the SD states. For example, the AMD calculations suggest that SD bands in $^{40}$Ca [10] and $^{44}$Ti [9] are constructed from triaxial shapes and their side bands exist. The triaxiality of superdeformations in this mass region has also been discussed in the three-dimensional coordinate-mesh Hartree-Fock (HF) [6]. Triaxiality is also an important feature of normal-deformed states in the A ∼ 40 region. Established triaxial states are in $^{40}$Ca, in which an observed K π = 2 + band with low excitation energies [2] is interpreted as a side band of a K π = 0 + band caused by the triaxial normal-deformation [5, 10, 15, 16]. Therefore, triaxiality is one of the key properties to clarify the structure of normal-deformations and superdeformations in A ∼ 40 nuclei. However, theoretical studies taking into account triaxiality are still limited due to the higher numerical costs than axially symmetric calculations. Quite recently, an SD K π = 0 + band built on the J π = 0 + state (2.12 MeV) in $^{40}$Ar was experimentally identified up to the J π = (12 $^{+}$) state by the $^{26}$Mg($^{18}$O, 2p2n)$^{40}$Ar reactions [17], which were the lightest SD states ever in N = Z nuclei. Superdeformations in N = Z nuclei in this mass region are frontier topics. To discuss triaxiality, features of unnatural spin-parity [ π = ( − ) J $^{+1}$] states are helpful data. In $^{40}$Ar, a state at 4.23 MeV was assigned to an unnatural parity J π = 3 + state by the $^{40}$Ar( α , α $^{′}$) reaction [18]. This J π = 3 + state decays to the J π = 2 + state in the SD band [19, 20]. The J π = 2 + state at 3.92 MeV just below the J π = 3 + at 4.23 MeV also decays to the SD band member J π = 0 $^{+}$, and the E 2 transition strength takes a non-negligible value [20]. These observations show that the states J π = 2 + and 3 + are candidates for members of the side band in the SD band, which can be discussed in relation to the triaxiality of the 2