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Ambiguous Thermometers

This puzzle is a bit different to my usual puzzles. The idea came about following a discussion in the Ctc fan discord server which somehow got onto the subject of thermometers, and whether they should be in Celsius or Fahrenheit.

Ambiguous Thermometers

    Each row, column and 3x3 box contains the digits 1-9.
    Each line between cells represents a 'thermometer'.
    Digits along a thermometer must increase either left to right or right to left.
    The digits on each thermometer when read from left to right represent a temperature either in degrees Celsius or Fahrenheit.
    No two thermometers show the same value but for every thermometer showing in °C there is another showing the same temperature in °F.
    In addition some cells have been marked with the letters C, F & K representing the value in that cell.
    Each letter represents the same number wherever it appears.