Assume the planted clique hypothesis. Fix an asymptotic sparse PCA testing sequence $(d_N,n_N,k_N,\theta_N)$ and suppose there is a randomized polynomial-time reduction from planted clique on $N$ vertices with clique size $\kappa_N=o(\sqrt N)$ to this sparse PCA sequence with the following two properties: under $G(N,1/2)$ the output distribution is within $o(1)$ total variation distance of the sparse PCA null, and under the planted-clique model the output distribution is within $o(1)$ total variation distance of the sparse PCA alternative with sparsity at most $k_N$ and spike strength at least $\theta_N$. Then no randomized polynomial-time sparse PCA test for this sequence can have type I plus worst-case type II error tending to $0$.