When people are taught multiple regression analysis in school, it typically is for across-respondent regression problems (pooled, aggregate models). Students are taught to look for predictor variables that are "significant" and to drop those that are "not significant".
However, with conjoint analysis problems, it's usually a different approach. Rather than running regressions across respondents (where one can often have 7x or more as many observations as parameters to estimate), we recognize the value of capturing heterogeneity (for application within market simulators) and instead run regressions within each respondent, where we typically have 3x or fewer observations relative to parameters.
Researchers usually do not hunt for parameters that globally are not significant in the OLS conjoint models, because conjoint attributes that may be insignificant for some respondents may be very significant for others. And, to pool the data with one large regression potentially would flag a coefficient as not significant when really there could be two groups of respondents who think it is very significant, but disagree about the sign of the relationship (so the opposing effects wash out if aggregated).
All this said, there is a very interesting stream of literature within the conjoint analysis community for CBC and Latent Class or HB analysis regarding Attribute Non-Attendance (ANA). Some researchers use latent class to detect groups of respondents who appear to be ignoring an attribute, then set the coefficients to zero within those groups. There are also opportunities for investigating ANA via HB. Both Latent Class and HB work with much more data than individual-level OLS in CVA to attempt to identify respondents who are ignoring certain attributes.