To determine the correct word to fill in the blank in the sentence, we need to consider the context and grammatical structure.

The sentence is: "Rosy and Nicky couldn’t provide any evidence to support _______ arguments."

We have four options:

• theirs
• there
• they're
• their

Now, let's analyze each option:

1. theirs: This is a possessive pronoun. It is used to indicate ownership and generally stands alone without a noun following it. Example: "Those books are theirs."

2. there: This is an adverb used to refer to a place or position. It is not suitable here as it doesn't indicate possession. Example: "The book is over there."

3. they're: This is a contraction of "they are." It doesn't fit the context as it's not showing possession. Example: "They're going to the park."

4. their: This is a possessive adjective used to indicate something belongs to "them." This fits the context of the sentence which requires a word indicating possession of the arguments.

The sentence contextually requires a possessive adjective to show ownership of the arguments by "Rosy and Nicky". Therefore, the correct option is:

their