capitalization - Should I capitalize the word "that" in a title?

Since the version of the word "that" might matter to the correct answer, I'll provide the actual title in question:

Title: Existing Solutions that Didn't Work

If someone could explain the why of the answer, I would be appreciative.

EDIT: It has been suggested that this is a duplicate, and that may be, in a sense. However, the after reading the suggested prior question, I'm no closer to my answer. I think it may have to do with my use of the word "that." Perhaps it is incorrect?

Using these definitions of the word "that:"
I'm having trouble figuring out which definition of that matches my title. Am I even using it correctly? Perhaps I should use which instead:

Possible Title: Existing Solutions which Didn't Work

If my original usage is incorrect, and I should use which, then the answer is simple since which is always a pronoun or adjective, and those are both capitalized in a title. However, I'm having difficulty understanding which part of speech that is in my original title.

Answer

As Edwin commented, whether or not you capitalize a pronoun in a title is a matter of style, however, the main style guides do not differ on this point.

The AP convention is to capitalize all words with four or more letters.

The APA Manual also capitalizes all words with four or more letters as well as all "major words." Only "conjunctions, articles, and short prepositions are not considered major words" (APA Manual, 6th Ed., p. 101).

The Chicago Manual of Style does not base headline-style capitalization on word length. It calls for capitalization of "nouns, pronouns, verbs, adjectives, and some conjunctions" (excluding the conjunctions and, but, for or and nor). Prepositions and articles are always lowercase unless the preposition is part of a verb phrase (like "Look Up").

In short, you're safe to capitalize the words that and which in your title as they are both relative pronouns.

As a side note: in American English, the convention would be to use the word that in your title, as it qualifies as a restrictive clause.