Complete intersections of quadrics and complete intersections on Segre varieties with common specializations

We investigate whether surfaces that are complete intersections of quadrics and complete intersection surfaces in the Segre embedded product ${\mathbf{P}}^1\times {\mathbf{P}}^k\hookrightarrow {\mathbf{P}}^{2k+1}$ can belong to the same Hilbert scheme. For $k=2$ there is a classical example; it comes from K3 surfaces in projective $5$-space that degenerate into a hypersurface on the Segre threefold. We show that for $k\ge 3$ there is only one more example. It turns out that its (connected) Hilbert scheme has at least two irreducible components. We investigate the corresponding local moduli problem.