Via $p$-adic Nevanlinna theory, we can deduce the algebraic independence of a certain collection of hypergeometric functions. Combining this result with a Siegel-Shidlovsky type theorem of Chirskii, we can deduce the algebraic independence of the values of these hypergeometric functions evaluated at linearly independent algebraic numbers. This is still work in progress, hence we will provide a high level overview of the result, together with the motivating background.