Jordan triple-system conjecture for the second Veronese power

Let $\operatorname{Jord}$ be the operad of Jordan algebras, and let $\operatorname{Jord}^{[2]}$ denote its second Veronese power. Let $\operatorname{JTS}$ be the operad of Jordan triple systems, whose defining operation is a trilinear operation satisfying symmetry in the first and third variables and the Jordan triple-system identity. The operation induced by a Jordan algebra is

$$(a_1a_2)a_3+a_1(a_2a_3)-a_2(a_1a_3).$$

Jordan triple-system conjecture. There is an isomorphism

$$\operatorname{JTS}\cong\operatorname{Jord}^{[2]}.$$

Equivalently, all relations satisfied by this operation in Jordan algebras follow from the axioms of Jordan triple systems; equivalently, the operad $\operatorname{Jord}^{[2]}$ is quadratic. A natural surjection $\operatorname{JTS}\to\operatorname{Jord}^{[2]}$ is known, but whether it is an isomorphism remains open.