By 'phrases as subjects', they probably mean cases like these:
Taking our Great Pyrenees for a walk wears me out.
Everything that looks like that is singular. For instance, this is
wrong:
Taking our dogs for walks wear me out.
Even though 'dogs' and 'walks' are obviously plural, the whole phrase is singular.
What you've got there ('rising inventories') is sometimes called a 'noun phrase', which might be the source of the confusion. Noun phrases just consist of a noun plus the modifiers that modify it. They can be either singular or plural, depending on whether the noun is singular or plural. Here, 'rising' isn't being used as a verb, it's being used as an adjective which modifies 'inventories'.
Interestingly, the following sentence would be
technically correct to the best of my knowledge, although it is a bit clunky and lacks concision.
The rising of inventories leads to production cutbacks.
Note the singular verb, 'leads'.