Hu–Li conjecture on the upper bound for $f(p)$

Let $p\ge 5$ be a prime, and let $f(p)$ be the smallest number such that every planar graph of girth $p$ with no cycles of lengths from $p+1$ through $f(p)$ admits a homomorphism to the cycle $C_p$. The Hu–Li conjecture.

$$f(p)\le p(p-2).$$

Equivalently, every planar graph of girth $p$ without cycles of lengths from $p+1$ to $p(p-2)$ is $C_p$-colorable. The paper proves finite lower and upper bounds for $f(p)$ but does not establish this proposed upper bound; it would imply Jaeger's conjecture that every planar graph of girth $2p-2$ has a homomorphism to $C_p$.