Existence conjecture for fullerene $c$-disks

A fullerene $c$-disk is a $(\{5,6,c\},3)$-sphere with exactly one $c$-gon. Let $p_6(c)$ denote the minimal possible number of $6$-gons in a fullerene $c$-disk, and let $v$ be the number of vertices.

Fullerene disk existence conjecture. Except for $(c,v)=(1,42)$, $(3,24)$, and $(5,22)$, a fullerene $c$-disk with $c\geq 1$ and $v$ vertices exists if and only if $v$ is even and

$$v\geq 2\bigl(p_6(c)+c+5\bigr).$$

The claim gives a complete parity-and-size criterion apart from three explicit exceptional pairs. The source provides no resolution of the conjecture.