Q. Could we have some prescriptions for sentence-initial “i.e.,” “e.g.,” “ibid.,” and the like in notes and parentheses? Capitalizing the first letter is widely felt to be awkward (see June Casagrande, “A Word, Please: A Guide to Using Latin Abbreviations, E.g. and I.e.,” December 12, 2015, in the Los Angeles Times), and at least one legal style guide prescribes lowercase “ibid.” at the beginning of a footnote (OSCOLA: Oxford University Standard for the Citation of Legal Authorities, 4th ed., Faculty of Law, 2012). Other style pundits have recommended periphrasis.
A. Chicago would apply an initial cap to each of those abbreviations at the beginning of a sentence, including at the beginning of a numbered footnote or endnote (Chicago styles these like sentences), where “ibid.” would often be the first word (see CMOS 14.34 for examples—and for why we prefer to use a shortened form of the source in favor of “ibid.”):
1. Ibid., 83.
We’d do the same for “i.e.” and “e.g.” (i.e., writing “I.e.” and “E.g.”). But those two abbreviations usually occur in parentheses in Chicago style, and there are no examples of either in the seventeenth edition of CMOS at the start of a sentence or a note. (Attentive readers will have noticed the initial caps for each in the title of the LA Times article in your question, which we decided were being used as nouns when we applied the rules for headline style at CMOS 8.159; “I.E.” and “E.G.” could also work, but we’d reserve that treatment for an all-caps heading. Cf. this Q&A on sp. and spp.)
We do make one exception that’s directly related to your question—for the abbreviation “p.” (page) in a parenthetical page citation at the end of a block quotation (which isn’t Latin but does occur in source citations). Because the “(p. 142)” at the end of the second block quotation in CMOS 13.70 follows a period, one might think that the p should be capitalized: “(P. 142).” But we decided that we prefer consistency for those lowercase p’s relative to other such parenthetical p’s—and that we don’t consider “(p. 142)” to work like a sentence in that context (though the word see may be implied before “p.”).