The total sense of hot or cold is set by the integration of the response of all the sensors in the skin, +- evenly distributed, involved in the experiment.
The sense of hot is proportional to the area and the sense of cold is proportional to the perimeter*$\delta r$ , for the same pressure (see the answer of user1355).
The outcome of the experiment can be described by the ratio hot/cold or _(area)/(perimeter*$\delta r$)_ $=\pi R^2/2 \delta r \pi R=R/2\delta r$, i.e proportional to $R$ .
Let Experiment open mouth have radius $10 R$ and Experiment almost closed mouth have radius $R$.
The ratio of the outcomes of the two experiments (open/closed) is 10 .
If someone has doubts, and I do have, that the perimeter can contribute significantly to a colder sensation than the overall outcome becames more contrasted (ratio of areas, proportional to $R^2$)
and the above example will give the value 100.
An array of thermometers can set the question, imo. One measure can be defined as the sum of all the temperatures.
The description in the question is correct, supported by experimentation (I did it ;), and have a physical and rational interpretation.