Let $G$ and $H$ be finite groups, generated by $s_1,\dotsc,s_d\in G$ and $t_1,\dotsc,t_d \in H$, respectively ($d\geq 1$).
Assume that each of the Cayley graphs $\operatorname{Cay}(G,\{s_1,\dotsc,s_d\})$ and $\operatorname{Cay}(H,\{t_1,\dotsc,t_d\})$ has spectral gap at least $\varepsilon>0$.
Assume that the set $U=\{(s_1,t_1),\dotsc,(s_d,t_d)\}$ generates the group $G\times H$.
Is there $c>0$, depending only on $\varepsilon$ and $d$, such that the spectral gap of $\operatorname{Cay}(G\times H, U)$ is at least $c$?
Above, the spectral gap of a connected graph $X$ is $d-\lambda$, where $\lambda$ is the second largest eigenvalue of the adjacency matrix of $X$.