New Mexico does have state constitutional preemption of firearms laws, so local governments may not restrict the possession or use of firearms.
In 1986, Article 2, Section 6 of the state constitution was amended to say, "No law shall abridge the right of the citizen to keep and bear arms for security and defense, for lawful hunting and recreational use and for other lawful purposes, but nothing herein shall be held to permit the carrying of concealed weapons. No municipality or county shall regulate, in any way, an incident of the right to keep and bear arms."
However, if this gets passed (albeit unconstitutional as it is), due to the unconstitutional nature of this bill it will be tied up in legal battles indefinitely, a long and expensive legal battle which the gun owners of Santa Fe or New Mexico can ill afford to pursue.
Be sure to add your voice to the petition, and take the time to email all of the city councilors listed in the linked list.