The moduli space of K3 surfaces forms a 20-dimensional family with countably many 19-dimensional components $M_d$ corresponding to the polarized K3s $(X,L)$ with $L^2=d$. The moduli space $M_d$ has a natural locus $M_d^W$ for each lattice $W$ with an embedding $W\subset H^{1,1}(X)\cap H^2(X,\mathbb Z)$. Is it known how these loci intersect?

For instance, is it true that $M_{d}^W\cap M_{e}\neq \emptyset$ for each $W,d,e$?

The moduli space of K3 surfaces forms a 20-dimensional family with countably many 19-dimensional components $M_d$ corresponding to the polarized K3s $(X,L)$ with $L^2=d$. The moduli space $M_d$ has a natural locus $M_d^W$ for each lattice $W$ with an embedding $W\subset H^{1,1}(X)\cap H^2(X,\mathbb Z)$. Is it known how these loci intersect?

For instance, is it true that $M_{d}^W\cap M_{e}\neq \emptyset$ for each $W,d,e$ provided $M_{d}^W\neq \emptyset$?