Iwahori-Hecke model for the universal supersingular representation

Let F be a non-archimedean local field with residue field Fq. When F is a finite extension of Qₚ, Anandavardhanan-Borisagar and Anandavardhanan-Jana introduced an Iwahori-Hecke model for the universal supersingular representation πᵣ in the regular case $ 0 < r < q-1$. When F=Qₚ, the first author introduced an Iwahori-Hecke model for πᵣ when $r = 0, p - 1$. We extend this result to an arbitrary local field F for $r = 0, q - 1$. We also write down an explicit non-split self-extension of πᵣ which has a four-dimensional space of $I(1)$-invariants when F = Qₚ and $r = 0, p-1$.