In [Russ. Math. Surv. 29, No. 2, 15--212 (1974; Zbl 0305.35014)] \textit{E. M. Landis} and \textit{O. A. Olejnik} proposed the following conjecture: If \(u(x, t)\) is a bounded solution of a uniformly parabolic equation \[ \sum_{i,j=1}^n \partial_i\big(a^{ij}(x)\partial_ju\big)+ b(x)\cdot\nabla u+c(x)u-\partial_t u=0\quad \text{in }\mathbb{R}^n\times[0,T], \] such that \[ |u(x,T)|\leq N e^{-|x|^{2+\varepsilon}},\quad x\in \mathbb{R}^n \] with positive constants \(N\) and \(\varepsilon\), then \(u(x,t)\equiv0\) in \(\mathbb{R}^n\times[0,T]\), provided that the coefficients of the equation satisfy appropriate conditions at infinity. The authors prove this conjecture for operators with space-time dependent smooth coefficients.