We consider the boundary dual of ${\it AdS}_3\times S^3\times K3$ for NS-flux $Q_5=1$, which is described by a sigma model with target space given by ${\rm Sym}^d(K3)/S_d$. Motivated by the CFT distance conjecture, we address the problem of deforming the singular orbifold ${\rm Sym}^d(K3)/S_d$ globally over the moduli space of the blow-up modulus. For this we use the language of ${\mathcal N}=4$ topological strings, which allows to make use of results in algebraic geometry, topological gravity and Hurwitz theory. This leads to a geometrization of the t' Hooft expansion to all orders in terms of "reduced" Gromow-Witten invariants. For this to work, some assumptions need to be made that we will discuss.