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In a family of experiments carried on all-metallic supercurrent nano-transistors a surprising gating effect has been recently shown. These include the full suppression of the critical supercurrent, the increase of quasiparticle population, the manipulation of the superconducting phase, and the broadening of the switching current distributions. Aside from the high potential for future applications, these findings raised fundamental questions on the origin of these phenomena. To date, two complementary hypotheses are under debate: an electrostatically-triggered orbital polarization at the superconductor surface, or the injection of highly-energetic quasiparticles extracted from the gate. Here, we tackle this crucial issue via a fully suspended gate-controlled Ti nano-transistor. Our geometry allows to eliminate any direct injection of quasiparticles through the substrate thereby making cold electron field emission through the vacuum the only possible charge transport mechanism. With the aid of a fully numerical 3D model in combination with the observed phenomenology and thermal considerations we can rule out, with any realistic likelihood, the occurrence of cold electron field emission. Excluding these two trivial phenomena is pivotal in light of understanding the microscopic nature of gating effect in superconducting nanostructures, which represents an unsolved puzzle in contemporary superconductivity. Yet, from the technological point of view, our suspended fabrication technique provides the enabling technology to implement a variety of applications and fundamental studies combining, for instance, superconductivity with nano-mechanics., 7 pages, 4 color figures