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We first quantize an action proposed by Casalbuoni and Gomis in 2014 that describes two massless relativistic scalar particles interacting via a conformally invariant potential. The spectrum is a continuum of massive states that may be interpreted as unparticles. We then obtain in a similar way the mass operator for a deformed action in which two terms are introduced that break the conformal symmetry: a mass term and an extra position-dependent coupling constant. A simple Ansatz for the latter leads to a mass operator with linear confinement in terms of an effective string tension <inline-formula><math display="inline"><semantics><mrow><mi>σ</mi><mspace width="0.166667em"></mspace></mrow></semantics></math></inline-formula>. The quantized model is confining when <inline-formula><math display="inline"><semantics><mrow><mi>σ</mi><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula> and its mass spectrum shows Regge trajectories. We propose a tensionless limit in which highly excited confined states reduce to (gapped) unparticles. Moreover, the low-lying confined bound states become massless in the latter limit as a sign of conformal symmetry restoration and the ratio between their masses and <inline-formula><math display="inline"><semantics><msqrt><mi>σ</mi></msqrt></semantics></math></inline-formula> stays constant. The originality of our approach is that it applies to both confining and conformal phases via an effective interacting model.