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We investigate the kination-amplified inflationary gravitational-wave background (GWB) interpretation of the signal recently reported by various pulsar timing array (PTA) experiments. Kination is a post-inflationary phase in the expansion history dominated by the kinetic energy of some scalar field, characterized by a stiff equation of state w = 1. Within the inflationary GWB model, we identify two modes that can fit the current data sets (NANOGrav and EPTA) with equal likelihood: the kination-amplification (KA) mode and the ordinary, no-kination-amplification (no-KA) mode. The multimodality of the likelihood motivates a Bayesian analysis with nested sampling. We analyze the free spectra of current PTA data and mock free spectra constructed with higher signal-to-noise ratios using nested sampling. The analysis of the mock spectrum designed to be consistent with the best fit to the NANOGrav 15 yr (NG15) data successfully reveals the expected bimodal posterior for the first time while excluding the reheating mode that appears in the fit to the current NG15 data, making a case for our correct and comprehensive treatment of potential multimodal posteriors arising from future PTA data sets. The resultant Bayes factor is <inline-formula> <mml:math><mml:mi>B</mml:mi><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>no</mml:mi><mml:mo>-</mml:mo><mml:mi>KA</mml:mi></mml:mrow></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mrow><mml:mi>KA</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>2.9</mml:mn><mml:mo>±</mml:mo><mml:mn>1.9</mml:mn></mml:math> </inline-formula>, indicating comparable statistical significance between the two modes. Given the theoretical model-building challenges of producing highly blue-tilted primordial tensor spectra, the KA mode has the advantage of requiring less blue primordial spectra, compared with the no-KA mode. The synergy between future cosmic microwave background polarization, pulsar timing, and laser interferometer measurements of gravitational waves will help resolve the ambiguity implied by the multimodal posterior in PTA-only searches.