Nanopores in 2D materials are highly desirable for DNA sequencing, yet achieving single-stranded DNA (ssDNA) transport through them is challenging. Using density functional theory calculations and molecular dynamics simulations we show that ssDNA transport through a pore in monolayer hexagonal boron nitride (h-BN) is marked by a basic nanomechanical conflict. It arises from the notably inhomogeneous flexural rigidity of ssDNA and causes high friction via transient DNA desorption costs exacerbated by solvation effects. For a similarly sized pore in bilayer h-BN, its self-passivated atomically smooth edge enables continuous ssDNA transport. Our findings shed light on the fundamental physics of biopolymer transport through pores in 2D materials.