The H-measure was proposed by Dr Hand in his article Measuring classifier performance: a coherent alternative to the area under the ROC curve (2009), as a replacement for AUC-ROC. I haven't been able to find an article anywhere about the strictly proper/proper property of this measure.
Is the H-measure a strictly proper or a proper scoring rule?
There is actually a 2022/23 follow-up paper by Hand and Anagnostopoulos, in which they answer this question (among others).
As such it is not really meaningful to ask whether a classification performance measure itself is strictly proper or not, but it is meaningful to ask whether the probability scoring rule on which it is based is strictly proper. For the H-measure, [...] the loss measure is strictly proper.
(Emphasis mine.)
