It just depends on the elements of the charges and the defenses but typically motive is not an element of a murder charge - that’s more for the investigation and probable cause to arrest than it is for the conviction. Apparently Luigi has been charged with second degree murder. In NY, a defendant charged with second degree murder can assert that he “acted under the influence of extreme emotional disturbance for which there was a reasonable explanation or excuse.”
That might potentially set up a basis for him to put forward that case - but I’m not sure, just because someone is disturbed enough to kill doesn’t mean it’s reasonable. It will depend on what the case law says about that defense. I strongly suspect that the state is going to want to avoid the issue altogether and will attempt to limit that evidence.
https://www.nysenate.gov/legislation/laws/PEN/125.25