Let \(A\) and \(B\) be centrally symmetric convex bodies in \(\mathbb{R}^ n(A=- A,\;B=-B)\). Does \(\text{vol}(A\cap L)<\text{vol}(B\cap L)\) for all \((n- 1)\)-dimensional subspaces \(L\) of \(\mathbb{R}^ n\) imply \(\text{vol}(A)<\text{vol}(B)\)? The answer for \(n=2\) is affirmative in a trivial way. A. Giannopoulos [Mathematika 37, No. 2, 239-244 (1990; Zbl 0696.52004)] proved that for \(n\geq 7\) the answer is negative for a special type of solid \(A_ 0\) and a ball \(B\). The author shows in the present paper that a solid \(B\) of type \(A_ 0\) and a small perturbation \(A\) of \(B\) give negative answer for dimension \(n=5,6\). Thus the Busemann- Petty problem is still open for \(n=3,4\).