Random walks on direct products of groups

Suppose \operatorname{SL}_{2}(\mathbb{F}_{p})\times \operatorname{SL}_{2}(\mathbb{F}_{p}) is generated by a symmetric set S of cardinality n where p is a prime number. Suppose the Cheeger constants of the Cayley graphs of \operatorname{SL}_{2}(\mathbb{F}_{p}) with respect to \pi_{L}(S) and \pi_{R}(S) are at least c_{0} , where \pi_{L} and \pi_{R} are the projections to the left and the right components of \operatorname{SL}_{2}(\mathbb{F}_{p})\times \operatorname{SL}_{2}(\mathbb{F}_{p}) , respectively. Then the Cheeger constant of the Cayley graph of \operatorname{SL}_{2}(\mathbb{F}_{p})\times \operatorname{SL}_{2}(\mathbb{F}_{p}) with respect to S is at least c where c is a positive number which only depends on n and c_{0} . This gives an affirmative answer to a question of Lindenstrauss and Varjú.