Inclusion conjecture for NN and BD generalized doubly nonnegative cones

Let $\mathbb{K}$ be a direct product of a nonnegative orthant and second-order cones. Denote by $\mathcal{DNN}_{\rm NN}(\mathbb{K})$ and $\mathcal{DNN}_{\rm BD}(\mathbb{K})$ the NN and BD generalized doubly nonnegative cones, respectively.

Inclusion conjecture.

$$\mathcal{DNN}_{\rm NN}(\mathbb{K}) \subseteq \mathcal{DNN}_{\rm BD}(\mathbb{K}).$$

The conjecture is motivated by the preceding theoretical observation and numerical experiment, which verify the inclusion for the specified setting and for the tested examples. A general proof of the stated inclusion is not provided.