Air Carrier First Officer reported inaccurate FICON resulted in poor to nil braking action during landing roll.

Narrative

Enroute from ZZZ to JAC we became aware of JAC being below minimums due to low visibility and +SN. We discussed divert options but elected to fly green dot enroute to allow weather to improve versus holding further from a divert option. Braking action reports on descent indicated that the runway was acceptable for landing as well as a reported 5/5/4 FICON but with no PIREPs. Per the software we used current conditions (Braking Action 4) and JAC short field data to compute our landing distance; showing distance was acceptable for landing based on reports. We discussed the need to touch down in the first 1;000 feet of the runway. Weather improved prior to descent and we commenced the approach. Winds were variable due to mountain conditions but were predominantly a direct cross of 15-20 knots with a slight quartering tail component until 200-300 feet. On about a mile final; I as the PF (Pilot Flying) made a small lateral maneuver on final to avoid a large flock of geese at our 11 o'clock position but otherwise a normal approach. Using visual references; I attempted to fly about a dot below glide path to have my aim point in the first 500 feet of touchdown. However; the shifting winds created a slight performance increase on short final and through the flare. The aircraft was configured and stable at touchdown and medium autobrakes were selected with the plan to begin max manual braking after touchdown. I felt that we had touched down within the prescribed first 800-1000 ft of the runway however data subsequently indicated touchdown logged at ~1300 ft Visual runway markings were not visible to us due to recent contamination and clearing operations were ongoing based on arrivals. I felt that a slight performance increasing wind gust ballooned my flare and I recall releasing back stick pressure to force the aircraft down but felt that my aim point in the first 500 ft would keep me within the 1000 ft touchdown requirement; so I felt comfortable with the small balloon and continued the landing. I began max manual braking along with selecting maximum reverse on the engines. I personally noted a slightly less than optimal deceleration but attributed it to the runway condition. We passed the A2 turnoff and continued slowing when I stated the runway felt like an ice rink and the aircraft was difficult to control with the cross winds. The aircraft just did not slow as expected. We transferred aircraft control when I felt the Captain's feet come on the rudder pedals and I transferred it with full or near full braking pressure. At some point around this time we received an alert. The Captain kept the brakes buried and the aircraft just skidded down the runway. We were already at a slow rate of speed; maybe 10-15 knots; but considering the braking pressure; I could tell we were on a sheet of ice. We slowed the aircraft to a stop on centerline and estimated we had 100- 200 feet of runway remaining at stop. I would rate actual braking action between poor and nil (Braking Action 2; at best) in the last third of the runway. Perhaps an overall FICON of 4/3/2. We exited the runway at the end and taxied to the gate; informing Tower of our estimated braking action as Braking Action Reports were in effect. Taxi in was uneventful. Tower however continued to provide the FICON as 5/5/4 with our PIREP of poor braking action after our report. When queried about the conflict of info; they stated they were required to report both. I think that may be a bad policy as their ATIS did not provide the PIREP; just the FICON on the Digital ATIS. Two other carriers elected to divert based / go missed based on our PIREP.Weather at JAC resulted in a contaminated runway surface; lack of accurate braking reports from large transport category aircraft; short runway; variable winds at the airport; and a slightly long landing due to a minor balloon and visual references being obscured.Might be worth double checking the FICON calibration of their equipment. Also; if thedeparture end of the runway is FICON limited; and there is a direct cross; consider swapping runways so that the end of the runway has the most favorable FICON.