An often-mentioned advantage of conjoint experiments over traditional survey experimental designs is that the former have the potential to mitigate social desirability bias. To what extent this is true may depend on a number of design choices -- a concern that has received surprisingly little empirical attention. I conducted two studies in which I randomly assigned respondents to three types of conjoint designs in order to manipulate their awareness to sensitive features and possibilities for justifying inappropriate answers (N = 7,059). The results show that design variations signicantly affect respondents' inferences about the research objective. However, there are no detectable differences between respondents' preferences across designs. This indicates that researchers using conjoint experiments should not compromise their choice of design to avoid social desirability bias.