Does the emission of semen with desire—without practicing masturbation—during a fast for expiation of unintentional killing invalidate the fast and necessitate making it up?

The expiation for killing is of three types, in sequence, and fasting for two consecutive months is one of them. If you were unable to free a slave and fasted for two months, then you performed an action that led to the emission of semen, your fast is invalidated and its continuity is broken due to your deliberate invalidation of it without an excuse, and what has passed becomes a supererogatory act. Continuity is only broken by an interruption between acts of worship done by choice. Therefore, it is obligatory to restart the fasting of the aforementioned two months. However, if what was emitted was madhy (pre-seminal fluid), there is a difference of opinion among scholars regarding the invalidation of the fast by it.