The equivalence of zero Jordan product determined and zero product determined algebras

Let $A$ be an $F$-algebra, possibly infinite dimensional. Write “zJpd” for zero Jordan product determined and “zpd” for zero product determined.

Equivalence conjecture. $A$ is zJpd if and only if it is zpd.

For finite-dimensional $F$-algebras, these notions coincide when the characteristic of $F$ is not $2$. The conjecture asks whether the equivalence holds for arbitrary, possibly infinite-dimensional, $F$-algebras.