"Care not," would have.

Someone who says they do not care whether they get a thing may still want it. The proverb contrasts an outward show of indifference with a wish to have what is being discussed.

The point is not that the person changes their mind later. Their noncommittal answer may leave a desire unspoken: they would accept the thing, even though they have not said so plainly.

Use it when someone responds to an offer with an indifferent-sounding answer, but would welcome the thing offered. It calls attention to the wish concealed by a reply such as “I don’t mind.”