Capitalization of the short form of a compound noun, with a definite article

Example:

Northern College’s senior administration is considering a $100 million investment in campus infrastructure. Alumni groups have objected, citing a record of poor stewardship of the College’s limited resources in the past. The administration contends that a review of similar spending among peer colleges in the Consortium for Greater Learning justifies the proposal.

Here, "Northern College" is the full proper noun, and there is no controversy that both words should be capitalized. However, I am less clear on whether subsequent mentions of this proper noun, and in particular in situations where the definite article form is used to refer to the proper noun ("the College") is correct.

It seems that, since the definite article form is a stand-in for a specific proper noun, capitalization is appropriate. But I don’t know if this is a style question, or is governed by specific rules. The subsequent use ("peer colleges") clearly does not refer to a specific proper noun, and so falls squarely into the non-capitalized category.

Other, similar questions here seem to indicate that this style of noun capitalization is common in legal writing, but perhaps is less common or appropriate outside of that context.