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We construct a family of Lagrangian tori $\Theta^n_s$ $\subset$$(\mathbb{C}P^1)^n$, $s \in (0,1)$, where $\Theta^n_{1/2} = \Theta^n$, is themonotone twist Lagrangian torus described by Chekanov-Schlenk. We show that for$n = 2m$ and $s \ge 1/2$ these tori are non-displaceable. Then by considering$\Theta^{k_1}_{s_1}$ $ \times$ $\cdots$ $\times$ $ \Theta^{k_l}_{s_l}$ $\times$ $ (S^2_{\mathrm{eq}})^{n - \sum_i k_i}$ $ \subset $$(\mathbb{C}P^1)^n$, with $s_i \in [1/2,1)$ and $k_i \in 2\mathbb{Z}_{>0}$,$\sum_i k_i \le n$ we get several $l$-dimensional families of non-displaceableLagrangian tori. We also show that there exists partial symplectic quasi-states$\zeta^{\mathfrak{b}_s}_{\textbf{e}_s}$ and linearly independent homogeneousCalabi quasimorphims $\mu^{\mathfrak{b}_s}_{\textbf{e}_s}$ or which$\Theta^{2m}_s$ are $\zeta^{\mathfrak{b}_s}_{\textbf{e}_s}$-superheavy and$\mu^{\mathfrak{b}_s}_{\textbf{e}_s}$-superheavy. We also prove a similarresult for $(\mathbb{C}P^2 3\bar{\mathbb{C}P^2}, \omega_\epsilon)$, where$\{\omega_\epsilon; 0 < \epsilon < 1\}$ is a family of symplectic forms in$\mathbb{C}P^2 3\bar{\mathbb{C}P^2}$, for which $\omega_{1/2}$ is monotone.

Continuum families of non-displaceable Lagrangian tori in $(\mathbb{C}P^1)^{2m}$