Linear maps on $\mathcal{L}(\ell_p^n,\ell_p^m)$, $(p\in \{1,\infty\})$ preserving parallel pairs

Two vectors $x,y$ of a Banach space are said to form a parallel (resp. triangle equality attaining or TEA) pair if $\|x+\lambda y\|=\|x\|+\|y\|$ holds for some scalar $\lambda$ with $|\lambda|=1$ (resp. $\lambda=1$). For $p\in \{1,\infty\},$ and $m,n\geq 2,$ we study the linear maps $T: \mathcal{L}(\ell_p^n, \ell_p^m) \to \mathcal{L}(\ell_p^n,\ell_p^m)$ that preserve parallel (resp. TEA) pairs, that is, those linear maps $T$ for which $T(A),T(B)$ form a parallel (resp. TEA) pair whenever $A,B$ form a parallel (resp. TEA) pair of $\mathcal{L}(\ell_p^n,\ell_p^m).$ We prove that if $T$ is non-zero, then the following are equivalent: (1) $T$ preserves TEA pairs. (2) $T$ preserves parallel pairs and rank$(T)>1$. (3) $T$ preserves parallel pairs and $T$ is invertible. (4) $T$ is a scalar multiple of an isometry.