Conjecture on canonical grading of general Artinian Gorenstein algebras

Let $k$ be a field of characteristic not equal to $2$, $3$, or $5$. For an algebra $A$, write $H_A(1)$ for the dimension of its degree-one component, and let its type be $(n,d)$ when $H_A(1)=n$ and the socle degree of $A$ is $d$. An algebra is canonically graded when it is isomorphic to its associated graded algebra.

Canonical-grading conjecture. A general algebra of type $(n,d)$ is canonically graded if and only if $(n,d)$ belongs to the following list:

$$\begin{array}{ll} d\leqslant 4 &\text{and } n\text{ arbitrary},\\ d=5 &\text{and } n\leqslant 6,\\ d=6 &\text{and } n=2,\\ d\text{ arbitrary} &\text{and } n=1. \end{array}$$

The conjecture identifies precisely when canonical grading is expected for a general algebra. The only-if direction is proved in the paper, while the converse is supported by computational evidence; the cases $d\leqslant 4$ follow from earlier work of Elias and Rossi.