Let $\Pi=(Gen_1,H_1)$ and $\Pi=(Gen_2,H_2)$ be two hash functions. Define $(Gen, H)$ so that $Gen$ runs $Gen_1$ and $Gen_2$ obtaining $s_1$ and $s_2$ respectively. Then let $H^{s_1,s_2}(x)=H^{s_1}(x)\|H^{s_2}(x)$, is it second pre-image resistant if at least one of $H^{s_1}$ and $H^{s_2}$ is second pre-image resistant?
I have found the paper Multicollisions in iterated functions, application to cascaded constructions.by Antoine Joux and it showed that the hash function $H^{s_1,s_2}(x)$ is not secure. But does this approach work for every hash function?