On the question itself, both my Chicago Manual of Style and Garner's Dictionary of Modern American Usage are silent.
THE CHICAGO MANUAL OF STYLE --- AND GRAMMAR
The Chicago Manual of Style (CMS) is an unparalleled resource for those engaged in publishing, particularly of academic material. But the Press decided to farm out the topic of grammar and usage, and the writer they selected was Bryan A. Garner, a former associate editor of the Texas Law Review who now teaches at Southern Methodist University School of Law and has written several popular books on usage and style. His chapter is unfortunately full of repetitions of stupidities of the past tradition in English grammar " more of them than you could shake a stick at.
FIVE MORE THOUGHTS ON THE THAT RULE
So what does Garner say in GMAU? He's an idiot. On page 832:
Suffice it to say here that if you see a which with neither a preposition nor a comma, dash, or parenthesis before it, it should probably be a that.
Twice, my aggressive truculence about the That Rule (and a collection of other zombie rules) has prompted editors to cave in to my craziness and let me do whatever I want. Me. Not anyone else, just me, for this one book. They were then baffled that I didn't view this response as really satisfactory. I pointed out that the scholarly books their firms published on English grammar uniformly failed to subscribe to the That Rule, so that their presses looked like packs of hypocrites and fools. They simply didn't get it. For them, one thing is scholarship, the other thing is practice. They're just different.
Every so often I really run off the rails and rant. Paraphrasing some from my e-mail to one of these presses:
Sometimes I wonder: if the people who make up style sheets and enforce them are so damn fond of arbitrary and indefensible "rules" not grounded in usage, even the usage of the intellectual elites, why don't they just invent some? Say, your press won't publish any word with the letter "z" in it, or any sentence that begins with a vowel letter, or any occurrence of the pseudocleft construction, or the sequence "is for" (no matter how it arises)? I can think of hundreds of entertaining "rules" of this sort. You could hire people to enforce them, and make every book published by your press ENTIRELY CONSISTENT with them. And then schoolchildren everywhere could be drilled on these "rules". Your press could go down in history.
Hey, John Dryden did it for stranded prepositions. Some still-unidentified person(s) did it for possessive antecedents for pronouns, less than a century ago. There's plenty of territory still available. Talk it up to your board.
Somewhat more seriously (though my rant is not entirely unserious), there are hundreds and hundreds of stylistic choices that could be excised. The option between that relatives and zero relatives, for example: the people (that) I met. The option between complementizer that and no complementizer, for another: I think (that) we should go. I could go on for quite a while. Why are we being allowed to make these choices willy-nilly? Why isn't there a CLEAR RULE about which choice to make? How is the That Rule different from these putative rules?