The conjecture on unobstructed contraction algebras and smooth centres

Let $A$ be a contraction algebra and let $d=\dim_{\Bbbk} A$. Denote by $\mathcal{M}_d$ the moduli space of associative unital algebras of dimension $d$, and by $[A]$ the point corresponding to $A$.

Unobstructedness and smooth-centre conjecture. The point $[A]$ is smooth in $\mathcal{M}_d$, equivalently, $A$ is unobstructed; moreover, the dimension of the centre is constant in an open neighbourhood of $[A]$, and the associated map

$$A\mapsto Z(A)$$

is smooth.

This conjecture proposes that contraction algebras have especially regular deformation theory: they are unobstructed, and their centres vary smoothly in families. It is motivated by the preceding result that contraction algebras of types $A$ and $D$ deform to a single semisimple algebra, but the supplied text gives no resolution of the conjecture.