Following a problem introduced by Schurch [M. Schurch, \({On\; the\; Depression\; of\; Graphs}\), Doctoral Dissertation, University of Victoria, 2013], we find exact values of the minimum number of colours required to properly edge colour \( K_n \), \( n \geq 6 \), using natural numbers, such that the length of a shortest maximal path of increasing edge labels is equal to three. This result improves the result of Breytenbach and Mynhardt [A. Breytenbach and C. M. Mynhardt, On the \(\varepsilon\)-to appear-Ascent Chromatic Index of Complete Graphs, \({Involve}\), to appear].