The ratio of two different noncentral $t$ density functions with the same number of degrees of freedom is strictly monotone, with sense depending on the relative values of the two noncentral constants.
William Kruskal. "The Monotonicity of the Ratio of Two Noncentral $t$ Density Functions." Ann. Math. Statist. 25 (1) 162 - 165, March, 1954. https://doi.org/10.1214/aoms/1177728857