Before revision: Recently, Chudnovsky, Spirkl and Zhong  proved that List 3-Colouring is polynomial-time solvable for (K_{1,s}^1,P_t)-free graphs for every t\geq 1 and s\geq 1.
After revision: Recently, Chudnovsky, Spirkl and Zhong (DM 2020) proved that List 3-Colouring is polynomial-time solvable for (K_{1,s}^1,P_t)-free graphs for every t\geq 1 and s\geq 1.