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Spin-induced deformations of compact binary components can be probed through their gravitational-wave signals and are characterized by a parameter called kappa (κ), which equals 1 for black holes. Measuring its symmetric combination, κs, enables tests of the black hole nature of binary constituents. While previous studies focused on circular binaries, we investigate eccentric systems and forecast the expected measurement precision using a Fisher matrix analysis. For a 10-solar-mass binary with component spins greater than 0.8, we find that the fractional error in κs decreases from about 18% (circular) to about 8% (4%) for initial eccentricities e0 = 0.2 (0.5) at 5 Hz with Cosmic Explorer. Compared to the Advanced LIGO design sensitivity, the combined effects of eccentricity and the improved sensitivity of CE enhance precision by nearly an order of magnitude.