Packing subdivisions into regular graphs

We show that, for any graph $F$ and $\eta >0$, there exists a $d_0=d_0(F,\eta )$ such that every $n$-vertex $d$-regular graph with $d\ge d_0$ has a collection of vertex-disjoint $F$-subdivisions covering at least $(1-\eta )n$ vertices. This verifies a conjecture of Verstraëte from 2002 and improves a recent result of Letzter, Methuku and Sudakov which additionally required $d$ to be at least polylogarithmic in $n$.