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We consider the operator in ${ℝ}^d,d\ge 2$, of the form $H={(-\Delta )}^l+V,l\>0$ with a function $V$ periodic with respect to a lattice in ${ℝ}^d$. We prove that the number of gaps in the spectrum of $H$ is finite if $8l\>d+3$. Previously the finiteness of the number of gaps was known for $4l\>d+1$. Various approaches to this problem are discussed.