The non-complete-intersection conjecture for formal semigroups of hyperbolic L-space knots

Let $K$ be a hyperbolic $L$--space knot, and suppose its formal semigroup $S_K$ is a semigroup with embedding dimension $e(S_K)\geq 4$. A semigroup is complete intersection when its semigroup algebra is a complete intersection.

Formal-semigroup conjecture. The formal semigroup $S_K$ is not a complete intersection semigroup.

For embedding dimension $3$, Herzog's theorem identifies complete intersection numerical semigroups with symmetric ones, and symmetry is necessary for a semigroup to be the formal semigroup of a knot. The conjecture proposes that the hyperbolic case behaves differently once the embedding dimension is at least $4$.