Consider that inner city schools are often massively over-funded per child to virtually no metrics benefit. My suggestion is to allocate some such (relatively minimal) funding earlier in life, in line with cognitive development, so that children may be more successful in school in spite of later funding. Ideally, reducing the later funding that is needed to help children achieve.
Whether you're aware of it or not, you're asking for research with a very specific set of variables.
Whereas I'm making a suggestion, as a Speech and Language Pathologist, which is in line with child development best practices that are predicated on more general child development research. The latter which has formulated a long standing set of child-development best practices. One of the most primary being that the child must hear as much language as possible, and much of it should come from reading stories to and with the child.
I'm unsure of your first statement as well. Either look for such a study yourself, or if one doesn't exist then one may need to be done including those very specific variables.
What I am sure of is that absence of household books guarantees a negative outcome. And therefore it would be considered to be unethical, in research or more generally, to keep households deprived of books. As doing so would both make success more or less impossible, generally, as well as more specifically deprive children who are cognitively dispositioned to pick up a book on their own despite their parents.