Conjectural strengthenings on curves without abelian points

Let $K$ be any number field and let $E$ be any elliptic curve over $K$. Write $K^{\operatorname{ab}}$ for the maximal abelian extension of $K$.

Conjectural strengthenings. The following statements hold:

a) There is a genus one curve $C$ over $K$, with Jacobian $E$, such that

$$C(K^{\operatorname{ab}})=\emptyset.$$

b) For all $d\geq 3$, there is a degree-$d$ plane curve $C$ over $\mathbb{Q}$ such that

$$C(K^{\operatorname{ab}})=\emptyset.$$

c) For all $g\geq 4$, there is a curve $C$ over $\mathbb{Q}$ of genus $g$ such that

$$C(K^{\operatorname{ab}})=\emptyset.$$

These are proposed as stronger statements relating the existence of curves with no points over maximal abelian extensions to genus, degree, and prescribed Jacobian. The source presents them as conjectural strengthenings rather than established results; their status is therefore open.