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Highly radio-loud quasars (HRLQs; log  R > 2.5) at z ≳ 4 show apparent enhanced X-ray emission compared to matched HRLQs at lower redshifts, perhaps due to a redshift-dependent fractional contribution to the X-ray luminosity from inverse-Compton scattering of cosmic microwave background photons (IC/CMB). Using new Chandra observations and archival X-ray data, we investigate this phenomenon with an optically flux-limited sample of 41 HRLQs at z  = 4–5.5 all with sensitive X-ray coverage, the largest sample utilized to date by a wide margin. X-ray enhancements are assessed using X-ray-to-optical flux ratios and spectral energy distributions. We confirm the presence of X-ray enhancements at a 4.9–5.3σ significance level, finding that the median factor of enhancement is ≈1.8 at our sample median redshift of z ≈ 4.4. Under a fractional IC/CMB model, the expected enhancement at lower redshifts is modest; e.g. ≈4 per cent at z ≈ 1.5. We also investigate a sample of seven radio-loud quasars (RLQs; log  R > 1) at even higher redshifts of z  = 5.6–6.8, using new and archival X-ray data. These RLQs also show evidence for X-ray enhancements by a median factor of ≈2.7 at a 3.7–4.9σ significance level. The X-ray spectral and other properties of these z  = 5.6–6.8 RLQs, however, pose challenges for a straightforward fractional IC/CMB interpretation of their enhancements. [ABSTRACT FROM AUTHOR]