Goto–Watanabe finite generation conjecture for symbolic Rees rings of space monomial curves

Let $R=k[x,y,z]$ be the polynomial ring in three variables over a field $k$, and let $P$ be the prime ideal defining the space monomial curve $(t^a,t^b,t^c)$ in $\mathbb A^3_k$. For a positive-characteristic field $k$, write

$$R_S(P)=\bigoplus_{m=0}^{\infty}P^{(m)}$$

for the symbolic Rees ring of $P$. Goto–Watanabe conjecture. The ring $R_S(P)$ is finitely generated.

In characteristic zero, symbolic Rees algebras for some such curves are known not to be finitely generated, whereas analogous examples in positive characteristic are known to be finitely generated. The general positive-characteristic case remains open.