A depth inequality for Boolean algebra ultraproducts at singular cardinals

Let $\boldsymbol{B}_i$ ($i<\kappa$) be Boolean algebras, let $D$ be an ultrafilter on $\kappa$, and let $\boldsymbol{B}=\prod_{i<\kappa}\boldsymbol{B}_i/D$. Write ${\rm Depth}(\boldsymbol{B})$ for the depth of a Boolean algebra and ${\rm \lim}_D(\langle {\rm Depth}(\boldsymbol{B}_i):i<\kappa\rangle)$ for the ultrafilter limit. Assume

$$\aleph_0={\rm cf}(\lambda)<\lambda,\qquad 2^{\aleph_0}<\lambda,$$ $$\kappa<\lambda,$$ $${\rm Depth}(\boldsymbol{B}_i)\leq\lambda\quad\text{for every }i<\kappa,$$

and

$$\lambda={\rm \lim}_D(\langle {\rm Depth}(\boldsymbol{B}_i):i<\kappa\rangle).$$

The depth inequality conjecture. In ZFC,

$${\rm Depth}(\boldsymbol{B})\leq\prod_{i<\kappa}{\rm Depth}(\boldsymbol{B}_i)/D.$$

This concerns whether the depth of an ultraproduct can exceed the ultraproduct of the depths in the singular-cardinal case of countable cofinality. The preceding theorem establishes the corresponding equality under $V=L$; the conjecture asks for the stated upper bound in ZFC when $2^{\aleph_0}<\lambda$.