Should be synonymous with "not like it as much."
E.g.
Do you prefer my alternative or ______ it?
Should be synonymous with "not like it as much."
E.g.
Do you prefer my alternative or ______ it?
Consider disdain as the 'opposite' of prefer:
disdain verb Consider to be unworthy of one's consideration. ‘By 1980, major record companies disdained the music, preferring the slicker stylings of the new wave.’ - ODO
Note the contrast between disdain and prefer in the selected example above. Disdain is arguably stronger as a negative term than prefer as a positive term, but it's also weaker than discard. It parallels prefer as a description of someone's attitude, as opposed to their action.
Your example sentence would be:
Note, however, that "Would you prefer X?" is often taken to contrast X with some sort of an equivalent for X. In that sense, the contrast isn't between a positive and a negative opinion about X. I think this arises from the nature of the word prefer. For example, "Would you prefer my soup?" implies an alternative such as my roast or her soup.
Note also that strictly-speaking, the word disdain doesn't take two 'arguments' for a comparison. In the dictionary's example, disdain applies to just "the music", whereas preferring has both "the music" and "the ... stylings" in view. Had the example been "... preferred the music, disdaining the ... stylings ...", preferred would still have had both in view, while disdaining would only apply to the "stylings". If you simply wish to compare two alternatives (say, yours and his), you can say the following without using an antonym to prefer:
Would dislike work? Or do you disfavor such a trivial answer? :p