Linear Preserver Problem (LPP) is a classical research area in matrix theory. Recently, LPP was extensively studied by relaxing or changing the linearity or bijectivity, and this paper follows such a way. Let \(\mathbb C\) be the field of complex numbers, \(M_n\) the space of all \(n\times n\) complex matrices, \(T_n\) the subset of \(M_n\) consisting of all upper-triangular matrices, \(\Gamma_n\) the subset of \(M_n\) consisting of all \(k\)-potent matrices (i.e., \(A\in M_n\) is said to be \(k\)-potent if \(A^k= A\)), \(T\Gamma_n\) the subset of \(\Gamma_n\) consisting of all upper-triangular matrices, and \(\text{GL}_n\) the general linear group consisting of all \(n\times n\) invertible matrices over \(\mathbb C\). The main result of the paper is the following: Let \(n\geq 3\) and \(\varphi:T_n\to M_n\) be a map such that \(A-\lambda B\in T\Gamma_n\) if and only if \(\varphi(A)- \lambda\varphi(B)\in \Gamma_n\), for every \(A,B\in T_n\) and \(\lambda\in\mathbb C\). Then there exist \(P\in \text{GL}_n\) and a scalar \(c\) with \(c^{k-1}=1\) such that either \(\varphi(A)= cPAP^{-1}\) for every \(A\in T_n\), or \(\varphi(A)= cPA^TP^{-1}\) for every \(A\in T_n\).