Categorical intermediate-Jacobian conjecture for prime Fano threefolds

Let $X$ and $X'$ be prime Fano threefolds of index one or two, with intermediate Jacobians $J(X)$ and $J(X')$ and Kuznetsov components $\mathcal{K}u(X)$ and $\mathcal{K}u(X')$. Categorical intermediate-Jacobian conjecture.

$$J(X)\cong J(X')\Longrightarrow\mathcal{K}u(X)\simeq\mathcal{K}u(X').$$

The forward implication from an appropriate Fourier–Mukai equivalence of Kuznetsov components to an isomorphism of intermediate Jacobians is known, while this converse is posed as a conjecture and remains open.