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The joint observation of core-collapse supernovae with gamma-ray bursts shows that jets can be launched in the aftermath of stellar core collapse, likely by a newly formed black hole that accretes matter from the star. Such gamma-ray bursts have only been observed accompanying Type Ibc supernovae, indicating a stellar progenitor that lost its hydrogen envelope before collapse. According to recent hypothesis, it is possible that jets are launched in core-collapse events even when the progenitors still retain their hydrogen envelopes; however, such jets are not able to burrow through the star and will be stalled into the interior of the progenitor star before escaping. These jets are called choked jets. High-energy neutrinos produced by such choked jets could escape the stellar envelope and could be observed. Here, we examine how multimessenger searches for high-energy neutrinos and core-collapse supernovae can detect or limit the fraction of stellar collapses that produce jets. We find that a high fraction of jet production is already limited by previous observational campaigns. We explore possibilities with future observations using Large Synoptic Survey Telescope, IceCube, and Km3NET.