I would say that your interpretation of my interpretation is correct. And consistent with the results you get.
An argument could be made that there is little need for Type I error control among these 9 comparisons: no mean is used in more than one comparison, and the overall test of interaction assesses whether all 9 differences are equal (versus the alternative that at least two differences are unequal).
If you wanted control then, as you note, you could save the SLICEDIFF output to a dataset using ODS and then use MULTTEST. You could also specify the 9 comparisons in a LSMESTIMATE statement with the ADJUST option.
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