Iwahori-Hecke model for the universal supersingular representation

Let $F$ be a non-archimedean local field with residue field ${\mathbb{F}}_q$. When $F$ is a finite extension of ${\mathbb{Q}}_p$, Anandavardhanan-Borisagar and Anandavardhanan-Jana introduced an Iwahori-Hecke model for the universal supersingular representation ${\pi }_r$ in the regular case $0<r<q-1$. When $F={\mathbb{Q}}_p$, the first author introduced an Iwahori-Hecke model for ${\pi }_r$ 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 ${\pi }_r$ which has a four-dimensional space of $I(1)$-invariants when $F={\mathbb{Q}}_p$ and $r=0,p-1$.