  A cube tiling of $\mathbb{R}^d$ is a family of pairwise disjoint cubes
$[0,1)^d+T=\{[0,1)^d+t:t\in T\}$ such that $\bigcup_{t\in
T}([0,1)^d+t)=\mathbb{R}^d$. Two cubes $[0,1)^d+t$, $[0,1)^d+s$ are called a
twin pair if $|t_j-s_j|=1$ for some $j\in [d]=\{1,\ldots, d\}$ and $t_i=s_i$
for every $i\in [d]\setminus \{j\}$. In $1930$, Keller conjectured that in
every cube tiling of $\mathbb{R}^d$ there is a twin pair. Keller's conjecture
is true for dimensions $d\leq 6$ and false for all dimensions $d\geq 8$. For
$d=7$ the conjecture is still open. Let $x\in \mathbb{R}^d$, $i\in [d]$, and
let $L(T,x,i)$ be the set of all $i$th coordinates $t_i$ of vectors $t\in T$
such that $([0,1)^d+t)\cap ([0,1]^d+x)\neq \emptyset$ and $t_i\leq x_i$. It is
known that if $|L(T,x,i)|\leq 2$ for some $x\in \mathbb{R}^7$ and every $i\in
[7]$ or $|L(T,x,i)|\geq 6$ for some $x\in \mathbb{R}^7$ and $i\in [7]$, then
Keller's conjecture is true for $d=7$. In the present paper we show that it is
also true for $d=7$ if $|L(T,x,i)|=5$ for some $x\in \mathbb{R}^7$ and $i\in
[7]$. Thus, if there is a counterexample to Keller's conjecture in dimension
seven, then $|L(T,x,i)|\in \{3,4\}$ for some $x\in \mathbb{R}^7$ and $i\in
[7]$.
