Let $D'$ be the set of degree $4$ smooth surfaces in $\mathbb{P}^3$, where for each $S\in D'$ there are 6 hyperplanes $H_1, \dots, H_6\subset\mathbb{P}^3$ in general positions, such that $S$ passes through all the intersection points of these planes, i. e. $(H_i\cap H_j\cap H_k)\in S$ for all $1\leq i < j < k \leq 6$. Let $D=\overline{D'}$, the closure of $D'$.
I think for any quadric surface $Q=z_1z_4-z_2z_3$, i.e. $\mathbb{P}^1\times\mathbb{P}^1$, that $Q^2\notin D$, where $Q^2$ is the scheme defined by $(z_1z_4-z_2z_3)^2=0$. But I don't have a rigorous proof, and hope to give $D$ a more tractable description.
Here I call an element in $D$ a Darboux surface because this is a generalization of Darboux curves (degree 4 Darboux curves are called Luroth quartics, which has been studied intensively). I think this is a classical subject, but didn't find any reference about it, so I guess maybe it has some other names? Thanks a lot!