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You misinterpret the result. Doing the experiment indoors assures
that the ice will be at the melting point. Pressure melting occurs
at the melting point at a relatively low pressure. A reasonable
way to test your hypothesis is to take wires of different thermal
conductivities and identical diameters (say stainless steel and
copper) and time their cutting. One would expect the times to
differ by the same proportion as their thermal conductivities if
your hypothesis is correct.
Leigh