What Prevents Resolving the Hubble Tension through Late-Time Expansion Modifications?

We demonstrate that Type Ia supernovae (SNe Ia) observations impose the critical constraint for resolving the Hubble tension through late-time expansion modifications. Applying the Fisher-bias optimization framework to cosmic chronometers (CC), baryon acoustic oscillations (BAO) from DESI DR2, Planck CMB, and Pantheon+ data, we find that: (i) deformations in $H(z \lesssim 3)$ (via $w(z)$ reconstruction) can reconcile tensions between CC, Planck, DESI BAO, and SH0ES measurements while maintaining or improving fit quality ($\Delta\chi^2 < 0$ relative to $\Lambda$CDM); (ii) In the neighborhood of Planck best-fit $\Lambda$CDM model, no cosmologically viable solutions targeting $H_0 \gtrsim 69$ satisfy SNe Ia constraints. MCMC validation confirms the maximum achievable $H_0 = 69.09\pm0.30$ ($\chi^2_{\rm BF} \approx \chi^2_{\Lambda\rm CDM}$) across all data combinations, indicating that the conflict between late-time $w(z)$ modifications and SNe Ia observations prevents complete resolution of the Hubble tension.