The conjecture that all knots are H-restricted

Let $K$ be a knot, and let $\mathcal{C}(K)$ be its complex of graded modules over the algebra $\mathcal{A}$. The knot is H-restricted when the non-acyclic indecomposable summands of $\mathcal{C}(K)$ are one $\mathcal{A}\{i\}$ for some $i$, together with one or several $C_1[j]\{k\}$ for $j,k\in\mathbb Z$. H-restrictedness conjecture. All knots are H-restricted. This property is known for $(2,2m+1)$-torus knots and the figure-eight knot, and it is preserved under connected sum; whether every knot has the property remains open.