Given $t$ distinct colors, the $t$ subsets of $t-1$ colors are ordered in some arbitrary manner. Let $(G_1, G_2, \ldots, G_t)$ be graphs. The $(t-1)$-chromatic Ramsey number, denoted by $r_{t-1}^t(G_1, G_2, \ldots, G_t)$, is defined to be the least number $n$ such that if the edges of the complete graph $K_n$ are colored in any fashion with $t$ colors, then for some $i$ the subgraph whose edges are colored with the $i$th subset of the colors contains $G_i$. In this paper, the authors determine the exact value of $r_4^5(G_1, G_2, \ldots, G_5)$ where each $G_i$ is a path. This verifies a conjecture of \textit{A. Khamseh} and \textit{G. R. Omidi} [Int. J. Comput. Math. 89, No. 10, 1303--1310 (2012; Zbl 1257.05095)].