Can anyone help me a problem in market design?

Consider a marriage market where matchings are formed using the DA women-proposing algorithm. However, there is an additional constraint that only matches among agents that have previously gone out on a date can be formed. You are asked to study the strategic form game where, before the DA algo-rithm is run, each man can ask exactly one woman on a date (no date request  is refused).
1) Is the strategy profile where each man asks for a date with the woman he would be matched in the men-optimal stable matching a Nash equi-librium?
2) Is any other strategy profile a Nash equilibrium of the game? Provide an example, or prove that there is no Nash equilibrium candidate other than the strategy profile in part 1)

Thanks in tons.