Flight Instructor reported an NMAC event during landing pattern training caused by a reckless maneuver of another aircraft in the landing pattern. The other pilot denied the event and refused to accept his reckless maneuver.

Narrative

I am a Flight Instructor FBO out of ZZZ1. We were conducting training out of our Cessna 172 Lima model. We were doing a routine practice flight followed by landings at ZZZ. We made a radio call roughly 7 miles out from the southeast that we were coming in for the left downwind of Runway XX. We were aware that the winds were not favoring that runway but because there were traffic already landing on XX; we decided to follow them in and re-assess the situation after we land. Roughly 60 seconds after making our radio call; we observed an airplane 800 ft. below us; coming in the opposite direction. At that time he seemed to be flying parallel with us. Upon him coming close to being abeam our left wing; the traffic suddenly made a sharp left turn unannounced and started ascending towards us. I made a split second decision to take evasive action and turn right and start a climb immediately. Thankfully he seemed to have passed less than 200 ft. and 100 ft. below us. I immediately asked on the radio if that traffic was on the frequency of ZZZ- and got no response after 3 attempts. We later proceeded to safely make a landing. After we taxied back to the runway; we observed that the same traffic was on an extended final practicing some sort of an ILS XX into ZZZ. We talked to him on the radio and he stated that 1. He did not hear us make the radio call which we definitely did. 2. He had us visual the entire time. I do not understand why if someone had us visual that they would turn direct towards us and would not listen out for radio calls in such a busy uncontrolled airport. His tail number is Aircraft Y - a twin piper Seminole. This is my first time making a report. I am currently a CFI/I and have been flying for the last 5 years. I am saying with full confidence this is the closest mid-air collision I have ever encountered and to see the negligence from the other aircraft is mind-blowing. For him to come out on ATC and state that he never heard us or had us visual the entire time made me confused even more. I am more than happy to provide more information and hope that proper action is taken. This is absolutely unacceptable.