Typical representations for Sp₄(F)

Let F be a non Archimedean local field with odd residual characteristic, and let K be a hyperspecial maximal compact subgroup of the p-adic symplectic group G=Sp₄(F). Let s be an inertial class for G in the Bernstein decomposition of the category of smooth representations of G, which is attached to a proper Levi subgroup L of G. We prove that the s-typical irreducible representations of K are the irreducible components of Ind_JₛK(λₛ), where (Jₛ,λₛ) is an s-type for G such that Jₛ⊂ K, and (Jₛ,λₛ) is a G-cover of a Bushnell-Kutzko maximal simple type for L.