This is a question about history. To my understanding the equivalence of intertial and gravitational mass for all known particles is a weird empirical coincidence that has been confirmed to high precision countless of times but doesn't really have a 'natural' reason. So it is to be expected that people in the past have speculated about yet unobserved particles for which the two masses are not equal and have tried to use these particles in order to explain yet unexplained, mass-related phenomena (like the observations that lead to the introduction of dark matter).
If people in (say) the 1930s have indeed tried this, then what did these theories look like (e. g. was the ratio gravitational/inertial mass for the hypothetical matter supposed to be greater or smaller than for ordinary matter?) and on which grounds have these theories eventually been rejected?