$2$-Local isometries of non-commutative Lorentz spaces

Let $\mathcal M $ be a von Neumann algebra equipped with a faithful normal finite trace $\tau$, and let $S\left( \mathcal{M}, \tau\right)$ be an $\ast $-algebra of all $\tau $-measurable operators affiliated with $\mathcal M $. For $x \in S\left( \mathcal{M}, \tau\right)$ the generalized singular value function $\mu(x):t\rightarrow \mu(t;x)$, $t>0$, is defined by the equality $\mu(t;x)=\inf\{\|xp\|_{\mathcal{M}}:\, p^2=p^*=p \in \mathcal{M}, \, \tau(\mathbf{1}-p)\leq t\}.$ Let $\psi$ be an increasing concave continuous function on $[0, \infty)$ with $\psi(0) = 0$, $\psi(\infty)=\infty$, and let $\Lambda_\psi(\mathcal M,\tau) = \left \{x \in S\left( \mathcal{M}, \tau\right): \ \| x \|_{\psi} =\int_0^{\infty}\mu(t;x)d\psi(t) < \infty \right \}$ be the non-commutative Lorentz space. A surjective (not necessarily linear) mapping $V:\, \Lambda_\psi(\mathcal M,\tau) \to \Lambda_\psi(\mathcal M,\tau)$ is called a surjective $2$-local isometry, if for any $x, y \in \Lambda_\psi(\mathcal M,\tau) $ there exists a surjective linear isometry $V_{x, y}:\, \Lambda_\psi(\mathcal M,\tau) \to \Lambda_\psi(\mathcal M,\tau)$ such that $V(x) = V_{x, y}(x)$ and $V(y) = V_{x, y}(y)$. It is proved that in the case when $\mathcal{M}$ is a factor, every surjective $2$-local isometry $V:\Lambda_\psi(\mathcal M,\tau) \to \Lambda_\psi(\mathcal M,\tau)$ is a linear isometry.