The wreath product construction for Jordan superalgebras and Poisson superalgebras

Let $J_1$ be a Jordan superalgebra with product $\circ$, let $D:J_1\to J_1$ be an odd superderivative such that $D^2=0$, and let $H$ be a Poisson superalgebra with products $(\,\{\,\ \})$. Define $J=H\otimes J_1$ with the product

$$(a\otimes x)\bullet(b\otimes y)=ab\otimes(x\circ y)+\{a,b\}\otimes D(x)\circ D(y)$$

Wreath product conjecture. The algebra $J$ is a Jordan superalgebra. In this case, $J$ should be called the wreath product of $J_1$ with $H$.

This proposes a wreath-product construction combining a Jordan superalgebra with a Poisson superalgebra, analogous to wreath products for groups, Lie algebras, and associative algebras. The supplied text presents the assertion as a question and gives no resolution.