The HeII transverse proximity effect -- enhanced HeII Ly$alpha$~transmission in a background sightline caused by the ionizing radiation of a foreground quasar -- offers a unique opportunity to probe the morphology of quasar-driven HeII reionization. We conduct a comprehensive spectroscopic survey to find $zsim3$ quasars in the foreground of 22 background quasar sightlines with HST/COS HeII Ly$alpha$~transmission spectra. With our two-tiered survey strategy, consisting of a deep pencil-beam survey and a shallow wide-field survey, we discover 131 new quasars, which we complement with known SDSS/BOSS quasars in our fields. Using a restricted sample of 66 foreground quasars with inferred HeII photoionization rates greater than the expected UV background at these redshifts ($Gamma_mathrm{QSO}^mathrm{HeII} > 5 times 10^{-16},mathrm{s}^{-1}$) we perform the first statistical analysis of the HeII transverse proximity effect. Our results show qualitative evidence for a large object-to-object variance: among the four foreground quasars with the highest $Gamma_mathrm{QSO}^mathrm{HeII}$ only one (previously known) quasar is associated with a significant HeII transmission spike. We perform a stacking analysis to average down these fluctuations, and detect an excess in the average HeII transmission near the foreground quasars at $3sigma$ significance. This statistical evidence for the transverse proximity effect is corroborated by a clear dependence of the signal strength on $Gamma_mathrm{QSO}^mathrm{HeII}$. Our detection places a purely geometrical lower limit on the quasar lifetime of $t_mathrm{Q} > 25,mathrm{Myr}$. Improved modeling would additionally constrain quasar obscuration and the mean free path of HeII-ionizing photons.
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