The genus-count formula for the exterior of K13n586

Let $B$ be the exterior of the knot $K13n586$, and let $\atil_B(n)=a_B(n+1)$ count connected essential surfaces of genus $n+1$. Let $\phi(n)$ denote Euler's totient function. Genus-count conjecture. One has

$$\atil_B(n)=\phi(n)\quad\text{for all }n>1.$$

This is based on computed data and the interpretation of connected surfaces as primitive lattice points; no proof or resolution is supplied.